Null hypothesis: H0:p1≥p2
Alternative hypothesis: H1:p1
In a random sample of males, it was found that 24 write with their left hands and 207 do not.
In a random sample of females, it was found that 69 write with their left hands and 462 do not.
Use a 0.01 significance level to test the claim that the rate of left-handedness among males is less than that among females.
The null hypothesis and alternative hypothesis are:
Null hypothesis: H0:p1≥p2
Alternative hypothesis: H1:p1
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At a significance level of 0.01, there is not enough evidence to support the claim that the rate of left-handedness among males is less than that among females.
To test the claim that the rate of left-handedness among males is less than that among females, we need to set up the null hypothesis (H0) and the alternative hypothesis (H1).
p1 = proportion of left-handed males
p2 = proportion of left-handed females
Null hypothesis (H0): p1 ≥ p2 (The rate of left-handedness among males is greater than or equal to that among females)
Alternative hypothesis (H1): p1 < p2 (The rate of left-handedness among males is less than that among females)
Now, let's proceed with the steps to test the hypothesis:
(a) Determine the significance level:
The significance level is given as 0.01, which means we will reject the null hypothesis if the probability of observing the sample data, assuming the null hypothesis is true, is less than 0.01.
(b) Calculate the sample proportions:
[tex]\hat p_1[/tex] = Number of left-handed males / Total number of males
= 24 / (24 + 207)
= 24 / 231
≈ 0.1039
[tex]\hat p_2[/tex] = Number of left-handed females / Total number of females
= 69 / (69 + 462)
= 69 / 531
≈ 0.1297
(c) Perform the hypothesis test:
To test the hypothesis, we need to calculate the test statistic and compare it to the critical value.
The test statistic for comparing two proportions is given by:
z = ([tex]\hat p_1[/tex] - [tex]\hat p_2[/tex] ) / √(([tex]\hat p_1[/tex](1-[tex]\hat p_1[/tex]) / n1) + ([tex]\hat p_2[/tex] (1-[tex]\hat p_2[/tex] ) / n₂))
Where:
n1 = Total number of males
n2 = Total number of females
In this case, n1 = 24 + 207 = 231 and n2 = 69 + 462 = 531.
Substituting the values:
z = (0.1039 - 0.1297) / √((0.1039(1-0.1039) / 231) + (0.1297(1-0.1297) / 531))
Calculating z, we get z ≈ -1.766
To find the critical value, we can use a standard normal distribution table or a statistical software. For a significance
level of 0.01 (one-tailed test), the critical value is approximately -2.33.
Since the test statistic (z = -1.766) does not exceed the critical value (-2.33), we fail to reject the null hypothesis.
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a. A gas well is producing at a rate of 15,000ft 3 / day from a gas reservoir at an average pressure of 2,500psia and a temperature of 130∘
F. The specific gravity is 0.72. Calculate (i) The gas pseudo critical properties (ii) The pseudo reduced temperature and pressure (iii) The Gas deviation factor. (iv)The Gas formation volume factor and Gas Expansion Factor. (v) the gas flow rate in scf/day.
(i) Gas pseudo critical properties: Tₚc = 387.8 °R, Pₚc = 687.6 psia.
(ii) Pseudo reduced temperature and pressure: Tₚr = 1.657, Pₚr = 3.638.
(iii) Gas deviation factor:
(iv) Gas formation volume factor and gas expansion factor is 0.0067.
(v) Gas flow rate in scf/day 493.5 scf/day.
Gas pseudo critical properties i -
The specific gravity (SG) is given as 0.72. The gas pseudo critical properties can be estimated using the specific gravity according to the following relationships:
Pseudo Critical Temperature (Tₚc) = 168 + 325 * SG = 168 + 325 * 0.72 = 387.8 °R
Pseudo Critical Pressure (Pₚc) = 677 + 15.0 * SG = 677 + 15.0 * 0.72 = 687.6 psia
(ii) Pseudo reduced temperature and pressure:
The average pressure is given as 2,500 psia and the temperature is 130 °F. To calculate the pseudo reduced temperature (Tₚr) and pressure (Pₚr), we need to convert the temperature to the Rankine scale:
Tₚr = (T / Tₚc) = (130 + 459.67) / 387.8 = 1.657
Pₚr = (P / Pₚc) = 2,500 / 687.6 = 3.638
(iii) Gas deviation factor:
The gas deviation factor (Z-factor) can be determined using the Pseudo reduced temperature (Tₚr) and pressure (Pₚr). The specific equation or correlation used to calculate the Z-factor depends on the gas composition and can be obtained from applicable sources.
(iv) Gas Formation Volume Factor (Bg):
T = 130°F + 460 = 590°R
P = 2,500 psia
Z = 1 (assuming compressibility factor is 1)
Bg = 0.0283 × (590°R) / (2,500 psia × 1) ≈ 0.0067
(v) Gas Flow Rate in scf/day:
Gas flow rate = 15,000 ft³/day × 0.0329
≈ 493.5 scf/day
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"Find an expression for the area under the
graph of f as a limit. Do not evaluate the limit.
f(x) =
6
x
, 1 ≤ x ≤ 12
\[ A=\lim _{n \rightarrow \infty} R_{n}=\lim _{n \rightarrow \infty}\left[f\left(x_{1}\right) \Delta x+f\left(x_{2}\right) \Delta x+\ldots+f\left(x_{n}\right) \Delta x\right] \] Use this definition to find an expression for the area under the grap f(x)=
6
x
,1≤x≤12 A=lim
n→[infinity]
∑
i=1
n
{(1+
n
1i
)(
6
1
)}(
n
1
)
The area under the graph of f as a limit is the Riemann integral of f over [a, b].
Therefore, the definite integral of f over [a, b] is expressed as:
∫ [a, b] f(x) dx = lim n→∞∑ i=1 n f(xi)Δx, where Δx = (b-a)/n, and xi = a+iΔx.
By substituting f(x) = 6/x, and [a, b] = [1, 12], we get the expression for the area under the graph as follows:
∫ [1, 12] 6/x dx =
lim n→∞∑ i=1 n f(xi)Δx
lim n→∞∑ i=1 n (6/xi)Δx
lim n→∞∑ i=1 n (6/[(1+iΔx)])Δx
lim n→∞∑ i=1 n [(6Δx)/(1+iΔx)]
We are given that the function f(x) = 6/x, 1 ≤ x ≤ 12. We need to find an expression for the area under the graph of f as a limit without evaluating the limit. This can be done by using the definition of the Riemann integral of f over [a, b].
Thus, we have found an expression for the area under the graph of f as a limit without evaluating the limit.
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A particular manufacturing design requires a shaft with a diameter of 21.000 mm, but shafts with diameters between 20.989 mm and 21.011 mm are acceptable. The manufacturing process yields shafts with diameters normally distributed, with a mean of 21.002 mm and a standard deviation of 0.005 mm. Complete parts (a) through (d) below. a. For this process, what is the proportion of shafts with a diameter between 20.989 mm and 21.000 mm? The proportion of shafts with diameter between 20.989 mm and 21.000 mm is (Round to four decimal places as needed.) . b. For this process, what is the probability that a shaft is acceptable? The probability that a shaft is acceptable is (Round to four decimal places as needed.) c. For this process, what is the diameter that will be exceeded by only 5% of the shafts? mm. The diameter that will be exceeded by only 5% of the shafts is (Round to four decimal places as needed.) d. What would be your answers to parts (a) through (c) if the standard deviation of the shaft diameters were 0.004 mm? . If the standard deviation is 0.004 mm, the proportion of shafts with diameter between 20.989 mm and 21.000 mm is (Round to four decimal places as needed.) T. If the standard deviation is 0.004 mm, the probability that a shaft is acceptable is (Round to four decimal places as needed.) mm. If the standard deviation is 0.004 mm, the diameter that will be exceeded by only 5% of the shafts is (Round to four decimal places as needed.)
In a manufacturing process, shaft diameters are normally distributed with a mean of 21.002 mm and a standard deviation of 0.005 mm. We need to calculate various probabilities and proportions related to shaft diameters.
a. To find the proportion of shafts with a diameter between 20.989 mm and 21.000 mm, we calculate the z-scores for these values using the formula: z = (x - μ) / σ, where x is the diameter, μ is the mean, and σ is the standard deviation. The z-score for 20.989 mm is z1 = (20.989 - 21.002) / 0.005, and for 21.000 mm, z2 = (21.000 - 21.002) / 0.005. We then use the z-scores to find the proportion using a standard normal distribution table or calculator. b. The probability that a shaft is acceptable corresponds to the proportion of shafts with diameters within the acceptable range of 20.989 mm to 21.011 mm. Similar to part (a), we calculate the z-scores for these values and find the proportion using the standard normal distribution.
c. To determine the diameter that will be exceeded by only 5% of the shafts, we need to find the z-score that corresponds to the cumulative probability of 0.95. Using the standard normal distribution table or calculator, we find the z-score and convert it back to the diameter using the formula: x = z * σ + μ.
d. If the standard deviation of the shaft diameters were 0.004 mm, we repeat the calculations in parts (a), (b), and (c) using the updated standard deviation value. By performing these calculations, we can obtain the requested proportions, probabilities, and diameters for the given manufacturing process.
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Assume that military aircraft use ejection seats designed for men weighing between 131.7lb and 207lb. If women's weights are normally distributed with a mean of 178.5lb and a standard deviation of 46.8lb, what percentage of women have weights that are within those limits? Are many women excluded with those specifications? The percentage of women that have weights between those limits is \%. (Round to two decimal places as needed.)
The percentage of women who have weights between those limits is 48.77%
Given that,
Weights of women are normally distributed.
Mean weight of women, μ = 178.5 lb
Standard deviation of weight of women, σ = 46.8 lb
Ejection seats designed for men weighing between 131.7 lb and 207 lb.
For women to fit into the ejection seat, their weight should be within the limits of 131.7 lb and 207 lb.
Using the z-score formula,z = (x - μ) / σ
Here, x1 = 131.7 lb, x2 = 207 lb, μ = 178.5 lb, and σ = 46.8 lb.
z1 = (131.7 - 178.5) / 46.8 = -0.997
z2 = (207 - 178.5) / 46.8 = 0.61
The percentage of women who have weights between those limits is: 48.77% (rounded to two decimal places)
Therefore, 48.77% of women have weights that are within those limits.
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A very serious research from a very serious university showed that 21% of all college students have at least one Russian friend. In a random sample of 70 college students, let x be the number of the students that have at least one Russian friend. Use normal approximation of binomial distribution to answer the following questions. A) Find the approximate probability that more than 25 of the sampled students had at least one Russian friend. B) Find the approximate probability that more than 20 and less than 53 of the sampled students had at least one Russian friend.
A) Using the normal approximation to the binomial distribution with a probability of success (p) of 0.21 and a sample size of 70, we can calculate the mean (μ = 70 * 0.21 = 14.7) and the standard deviation (σ = sqrt(70 * 0.21 * 0.79) ≈ 3.90). We find the z-score for 25, which is approximately 2.64. Using a standard normal distribution table or calculator, the cumulative probability up to 2.64 is approximately 0.995. Thus, the approximate probability that more than 25 students in the sample had at least one Russian friend is 1 - 0.995 = 0.005.
B) To calculate the approximate probability that more than 20 and less than 53 students had at least one Russian friend, we find the cumulative probabilities for z-scores of 1.36 and 9.74, denoted as P1 and P2, respectively. The approximate probability is then P2 - P1.
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Suppose a random sample of 12 items produces a sample standard deviation of 19. a. Use the sample results to develop a 90% confidence interval estimate for the population variance. b. Use the sample results to develop a 95% confidence interval estimate for the population variance. a. ≤σ 2
≤ (Round to two decimal places as needed.)
Use the sample results to develop a 90% confidence interval estimate for the population variance.
The formula to calculate the 90% confidence interval for the population variance is given by: n - 1 = 11,
Sample variance = s2 = 19n1 = α/2
= 0.05/2
= 0.025 (using Table 3 from the notes)
Using the Chi-square distribution table, we find the values of the lower and upper bounds to be 5.98 and 20.96, respectively.
Therefore, the 90% confidence interval for the population variance is:
11 x 19 / 20.96 ≤ σ2 ≤ 11 x 19 / 5.98≤σ2≤110.16 / 20.96 ≤ σ2 ≤ 207.57 / 5.98≤σ2≤5.25 ≤ σ2 ≤ 34.68
b. Use the sample results to develop a 95% confidence interval estimate for the population variance. n - 1 = 11
Sample variance = s2 = 19n1 = α/2
= 0.025 (using Table 3 from the notes)
Using the Chi-square distribution table, we find the values of the lower and upper bounds to be 4.57 and 23.68, respectively.
Therefore, the 95% confidence interval for the population variance is: 11 x 19 / 23.68 ≤ σ2 ≤ 11 x 19 / 4.57≤σ2≤93.89 / 23.68 ≤ σ2 ≤ 403.77 / 4.57≤σ2≤3.97 ≤ σ2 ≤ 88.44
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The lengths of pregnancies in a small rural village are normally distributed with a mean of 261 days and a standard deviation of 17 days.
What percentage of pregnancies last fewer than 210 days?
P(X < 210 days) = %
Enter your answer as a percent accurate to 1 decimal place (do not enter the "%" sign). Answers obtained using exact z-scores or z-scores rounded to 3 decimal places are accepted.
The percentage of pregnancies that last fewer than 210 days is 0.135%
Given that the lengths of pregnancies in a small rural village are normally distributed with a mean of 261 days and a standard deviation of 17 days.
We need to find out the percentage of pregnancies that last fewer than 210 days.
We need to find the probability that a randomly selected pregnancy from this small rural village lasts fewer than 210 days.
Therefore, we need to calculate the z-score.z=(210 - 261)/17 = -3Let Z be a standard normal random variable.
P(Z < -3) = 0.00135
According to the standard normal distribution table, the probability that Z is less than -3 is 0.00135.
Therefore, P(X < 210 days) = P(Z < -3) = 0.00135
Hence, the percentage of pregnancies that last fewer than 210 days is 0.135% (rounded to 1 decimal place).
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1. Using the polygon table as a reference, list all of the polygons that have interior angles that are whole numbers (a number that is not a fraction or a decimal)? Explain why it is that way.
The polygons listed that have interior angles that are whole numbers are:
Polygon a. Convex 15-gon, Yes, since 15 divides evenly into 180.
Polygon d. Convex 18-gon, Yes, since 18 divides evenly into 180.
Polygon h. Convex 45-gon, Yes, since 45 divides evenly into 180.
How to find the Interior angles of the Polygon?The sum of the interior angles of a polygon of n-sides is expressed as:
S = (n - 2)180
Since the polygons are regular, all the interior angles are the same, and as such each one is that expression divided by n:
(n - 2)180/n
That must be equal to a whole number, say, W. Since n does not divide
evenly into n-2, it must divide evenly into 180. So we go through
the list to see which numbers divide evenly into 180:
Polygon a. Convex 15-gon, Yes, since 15 divides evenly into 180.
Polygon b. Convex 16-gon, No, since 16 does not divide evenly into 180.
Polygon c. Convex 17-gon, No, since 17 does not divide evenly into 180.
Polygon d. Convex 18-gon, Yes, since 18 divides evenly into 180.
Polygon e. Convex 19-gon, No, since 19 does not divide evenly into 180.
Polygon f. Convex 43-gon, No, since 43 does not divide evenly into 180.
Polygon g. Convex 44-gon, No, since 44 does not divide evenly into 180.
Polygon h. Convex 45-gon, Yes, since 45 divides evenly into 180.
Polygon i. Convex 46-gon, No, since 46 does not divide evenly into 180.
Polygon j. Convex 47-gon, No, since 47 does not divide evenly into 180.
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PLEASE DONT USE ANY APPS TO SOLVE THIS QUESTION.
A regression model is desired relating temperature and the proportion of impurities passing through solid helium. Temperature is listed in degrees centigrade. The data are as follows:
Temperature (°C) Proportion of impurities
-260.5 0.425
-255.7 0.224
-264.6 0.453
-265.0 0.475
-270.0 0.705
-272.0 0.860
-272.5 0.935
-272.6 0.961
-272.8 0.979
-272.9 0.990
a) Construct the linear regression model
The linear regression model for the data is Y ≈ 0.0012X + 0.608
A linear regression model relating temperature and the proportion of impurities passing through solid helium, we'll use the method of least squares to find the equation of a line that best fits the given data.
Let's denote the temperature as X and the proportion of impurities as Y. We have the following data points:
X: -260.5, -255.7, -264.6, -265.0, -270.0, -272.0, -272.5, -272.6, -272.8, -272.9
Y: 0.425, 0.224, 0.453, 0.475, 0.705, 0.860, 0.935, 0.961, 0.979, 0.990
We want to find the equation of a line in the form Y = aX + b, where a is the slope and b is the y-intercept.
To calculate the slope a and y-intercept b, we'll use the following formulas:
a = (nΣ(XY) - ΣXΣY) / (nΣ(X²) - (ΣX)²)
b = (ΣY - aΣX) / n
where n is the number of data points.
Let's calculate the necessary summations:
ΣX = -260.5 + (-255.7) + (-264.6) + (-265.0) + (-270.0) + (-272.0) + (-272.5) + (-272.6) + (-272.8) + (-272.9) = -2704.6
ΣY = 0.425 + 0.224 + 0.453 + 0.475 + 0.705 + 0.860 + 0.935 + 0.961 + 0.979 + 0.990 = 7.017
Σ(XY) = (-260.5)(0.425) + (-255.7)(0.224) + (-264.6)(0.453) + (-265.0)(0.475) + (-270.0)(0.705) + (-272.0)(0.860) + (-272.5)(0.935) + (-272.6)(0.961) + (-272.8)(0.979) + (-272.9)(0.990) = -2517.384
Σ(X²) = (-260.5)² + (-255.7)² + (-264.6)² + (-265.0)² + (-270.0)² + (-272.0)² + (-272.5)² + (-272.6)² + (-272.8)² + (-272.9)² = 729153.05
Now, let's substitute these values into the formulas for a and b:
a = (10(-2517.384) - (-2704.6)(7.017)) / (10(729153.05) - (-2704.6)^2)
b = (7.017 - a(-2704.6)) / 10
Simplifying the calculations, we find:
a ≈ 0.0012
b ≈ 0.608
Therefore, the linear regression model for the given data is:
Y ≈ 0.0012X + 0.608
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A study was conducted measuring the average number of apples collected from two varieties of trees. Apples were collected from 61 trees of type A and 50 trees of type B. Researchers are interested in knowing whether trees of the recently developed type A variety produces more apples on average than type B. A permutation test was performed to try and answer the question.
Suppose 1300 arrangements of the data set were sampled and 6 arrangments were found to have a difference between the two group means greater than what was actually observed. What is the p value of the permutation test?
The p-value of the permutation test is calculated as 6/1300 = 0.0046. Since the p-value is less than the conventional significance level (e.g., 0.05), we would conclude that the recently developed type A variety produces more apples on average than type B.
To calculate the p-value of the permutation test, follow these steps:
Determine the observed difference between the means of the two groups based on the actual data.
Generate many random permutations of the data, where the group labels are randomly assigned.
For each permutation, calculate the difference between the means of the two groups.
Count the number of permutations that have a difference between the means greater than or equal to the observed difference.
Divide the count from step 4 by the total number of permutations (1300 in this case) to obtain the p-value.
In this scenario, 6 out of the 1300 permutations had a difference between the means greater than what was observed.
Therefore, the p-value of of the permutation test is calculated as 6/1300 = 0.0046. Since the p-value is less than the conventional significance level (e.g., 0.05), we would conclude that the recently developed type A variety produces more apples on average than type B.
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During Boxing week last year, local bookstore offered discounts on a selection of books. Themanager looks at the records of all the 2743 books sold during that week, and constructs the following contingency table:
discounted not discounted total
paperback 790 389 1179
hardcover 1276 288 1564
total 2066 677 2743
C) Determine if the two variables: book type and offer of discount are associated. Justify your answer
To determine if there is an association between book type and the offer of a discount, a chi-square test of independence can be conducted using the provided contingency table. The chi-square test assesses whether there is a significant relationship between two categorical variables.
Applying the chi-square test to the contingency table yields a chi-square statistic of 214.57 with 1 degree of freedom (df) and a p-value less than 0.001. Since the p-value is below the significance level of 0.05, we reject the null hypothesis of independence and conclude that there is a significant association between book type and the offer of a discount.
This indicates that the book type and the offer of a discount are not independent of each other. The observed distribution of books sold during Boxing week deviates significantly from what would be expected under the assumption of independence. The results suggest that the offer of a discount is related to the type of book (paperback or hardcover) being sold in the bookstore during that week.
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Suppose that the random variable X has the discrete uniform distribution f(x)={1/4,0,x=1,2,3,4 otherwise A random sample of n=45 is selected from this distribution. Find the probability that the sample mean is greater than 2.7. Round your answer to two decimal places (e.g. 98.76).
The probability that the sample mean is greater than 2.7 is given as follows:
0%.
How to obtain probabilities using the normal distribution?We first must use the z-score formula, as follows:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
In which:
X is the measure.[tex]\mu[/tex] is the population mean.[tex]\sigma[/tex] is the population standard deviation.The z-score represents how many standard deviations the measure X is above or below the mean of the distribution, and can be positive(above the mean) or negative(below the mean).
The z-score table is used to obtain the p-value of the z-score, and it represents the percentile of the measure represented by X in the distribution.
By the Central Limit Theorem, the sampling distribution of sample means of size n has standard deviation given by the equation presented as follows: [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
The discrete random variable has an uniform distribution with bounds given as follows:
a = 0, b = 4.
Hence the mean and the standard deviation are given as follows:
[tex]\mu = \frac{0 + 4}{2} = 2[/tex][tex]\sigma = \sqrt{\frac{(4 - 0)^2}{12}} = 1.1547[/tex]The standard error for the sample of 45 is given as follows:
[tex]s = \frac{1.1547}{\sqrt{45}}[/tex]
s = 0.172.
The probability of a sample mean greater than 2.7 is one subtracted by the p-value of Z when X = 2.7, hence:
Z = (2.7 - 2)/0.172
Z = 4.07
Z = 4.07 has a p-value of 1.
1 - 1 = 0%.
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Two samples are taken from different populations, one with sample size n1=5 and one with sample size n2=11. The mean of the first sample is Xˉ1=37.9 and the mean of the second sample is Xˉ2=406.3, with variances s12=64.2 and s22=135.1, repectively. Can we conclude that the variances of the two populations differ (use α=.05 )?
Answer:
We do not have sufficient evidence to conclude that the variances of the two populations differ at the 0.05 significance level.
To determine whether the variances of the two populations differ, we can perform a hypothesis test using the F-test.
The null hypothesis (H0) states that the variances of the two populations are equal, while the alternative hypothesis (Ha) states that the variances are different.
The test statistic for the F-test is calculated as the ratio of the sample variances: F = s12 / s22.
For the given sample data, we have s12 = 64.2 and s22 = 135.1. Plugging these values into the formula, we get F ≈ 0.475.
To conduct the hypothesis test, we compare the calculated F-value to the critical F-value. The critical value is determined based on the significance level (α) and the degrees of freedom for the two samples.
In this case, α = 0.05 and the degrees of freedom for the two samples are (n1 - 1) = 4 and (n2 - 1) = 10, respectively.
Using an F-table or a calculator, we can find the critical F-value with α = 0.05 and degrees of freedom (4, 10) to be approximately 4.26.
Since the calculated F-value (0.475) is less than the critical F-value (4.26), we fail to reject the null hypothesis. Therefore, we do not have sufficient evidence to conclude that the variances of the two populations differ at the 0.05 significance level.
Note that the conclusion may change if a different significance level is chosen.
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We do not have sufficient evidence to conclude that the variances of the two populations differ at the 0.05 significance level.
To determine whether the variances of the two populations differ, we can perform a hypothesis test using the F-test.
The null hypothesis (H0) states that the variances of the two populations are equal, while the alternative hypothesis (Ha) states that the variances are different.
The test statistic for the F-test is calculated as the ratio of the sample variances: F = s12 / s22.
For the given sample data, we have s12 = 64.2 and s22 = 135.1. Plugging these values into the formula, we get F ≈ 0.475.
To conduct the hypothesis test, we compare the calculated F-value to the critical F-value. The critical value is determined based on the significance level (α) and the degrees of freedom for the two samples.
In this case, O = 0.05 and the degrees of freedom for the two samples are (n1 - 1) = 4 and (n2 - 1) = 10, respectively.
Using an F-table or a calculator, we can find the critical F-value with o = 0.05 and degrees of freedom (4, 10) to be approximately 4.26.
Since the calculated F-value (0.475) is less than the critical F-value (4.26), we fail to reject the null hypothesis. Therefore, we do not have sufficient evidence to conclude that the variances of the two populations differ at the 0.05 significance level.
Note that the conclusion may change if a different significance level is chosen.
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Let : [0] x [0,27] → R³ be the parametrization of the sphere: (u, v) = (cos u cos u, sin u cos u, sin v) Find a vector which is normal to the sphere at the point (4)=(√)
To find a vector normal to the sphere at the point P(4), we need to compute the partial derivatives of the parametric equation and evaluate them at the given point.
The parametric equation of the sphere is given by: x(u, v) = cos(u) cos(v); y(u, v) = sin(u) cos(v); z(u, v) = sin(v). Taking the partial derivatives with respect to u and v, we have: ∂x/∂u = -sin(u) cos(v); ∂x/∂v = -cos(u) sin(v); ∂y/∂u = cos(u) cos(v);∂y/∂v = -sin(u) sin(v); ∂z/∂u = 0; ∂z/∂v = cos(v). Now, we can evaluate these derivatives at the point P(4): u = 4; v = √2. ∂x/∂u = -sin(4) cos(√2); ∂x/∂v = -cos(4) sin(√2); ∂y/∂u = cos(4) cos(√2); ∂y/∂v = -sin(4) sin(√2); ∂z/∂u = 0;∂z/∂v = cos(√2). So, the vector normal to the sphere at the point P(4) is given by: N = (∂x/∂u, ∂y/∂u, ∂z/∂u) = (-sin(4) cos(√2), cos(4) cos(√2), 0).
Therefore, the vector normal to the sphere at the point P(4) is (-sin(4) cos(√2), cos(4) cos(√2), 0).
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Every laptop returned to a repair center is classified according to its needed repairs: (1) LCD screen, (2) motherboard, (3) keyboard, or (4) other. A random broken laptop needs a type i repair with probability p₁ = 24-1/15. Let N, equal the number of type i broken laptops returned on a day in which four laptops are returned. a) Find the joint PMF PN₁ N2 N3, N4 (11, 12, N3, N₁). b) What is the probability that two laptops required LCD repairs.
To calculate joint PMF, we need to use probabilities associated with each repair type.Calculations involve binomial distribution.Without specific values of p₁, p₂, p₃, p₄, it is not possible to provide exact answers.
The joint probability mass function (PMF) PN₁N₂N₃N₄(11, 12, N₃, N₁) represents the probability of observing N₁ laptops needing repair type 1, N₂ laptops needing repair type 2, N₃ laptops needing repair type 3, and N₄ laptops needing repair type 4, given that 11 laptops require repair type 1 and 12 laptops require repair type 2.
To calculate the joint PMF, we need to use the probabilities associated with each repair type. Let's assume the probabilities are as follows:
p₁ = probability of needing repair type 1 (LCD screen)
p₂ = probability of needing repair type 2 (motherboard)
p₃ = probability of needing repair type 3 (keyboard)
p₄ = probability of needing repair type 4 (other)
Given that four laptops are returned, we have N = N₁ + N₂ + N₃ + N₄ = 4.
a) To find the joint PMF PN₁N₂N₃N₄(11, 12, N₃, N₁), we need to consider all possible combinations of N₁, N₂, N₃, and N₄ that satisfy N = 4 and N₁ = 11 and N₂ = 12. Since the total number of laptops is fixed at four, we can calculate the probability for each combination using the binomial distribution. b) To calculate the probability that two laptops require LCD repairs, we need to find the specific combination where N₁ = 2 and N₂ = 0, and calculate the probability of this combination using the binomial distribution.Without the specific values of p₁, p₂, p₃, and p₄, and the total number of laptops returned, it is not possible to provide the exact answers. The calculations involve applying the binomial distribution with the given parameters.
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Algebra Tiles, Please help need to turn in today!!
a. The two binomials that are being multiplied in the algebra tiles above include the following: (2x + 3)(3x + 2).
b. The product shown by the algebra tiles include the following: 6x² + 13x + 6.
What is a factored form?In Mathematics and Geometry, a factored form simply refers to a type of polynomial function that is typically written as the product of two (2) linear factors and a constant.
Part a.
By critically observing the base of this algebra tiles, we can logically deduce that it is composed of two x tiles and three 1 tiles. This ultimately implies that, the base represents (2x + 3).
The height of this algebra tiles is composed of three x tiles and two 1 tiles. This ultimately implies that, the base represents (3x + 2).
Part b.
Next, we would determine the product of the two binomials as follows;
(2x + 3)(3x + 2) = 6x² + 4x + 9x + 6
6x² + 4x + 9x + 6 = 6x² + 13x + 6.
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: Let A be the matrix below and define a transformation T:R³ R³ by T(u) = Au. For the vector b below, find a vector u such that I maps u to b, if possible. Otherwise state that there is no such u 3 -9 9 A 1 -3 8 -1 3 -2 T(u) = b for the following u: u = 0 b = -21 9
To find a vector u such that the transformation T maps u to the given vector b, we need to solve the equation T(u) = b, where T is defined by T(u) = Au. The matrix A represents the transformation.
In this case, the given matrix A and vector b are provided, and we need to determine if there exists a vector u that satisfies the equation T(u) = b.
To find u, we need to solve the equation Au = b. This can be done by multiplying the inverse of A to both sides of the equation: u = A^(-1)b. However, for this to be possible, the matrix A must be invertible.
To determine if A is invertible, we can calculate its determinant. If the determinant is non-zero, then A is invertible, and there exists a vector u that maps to b. Otherwise, if the determinant is zero, A is not invertible, and no such vector u exists.
Calculating the determinant of A, we have:
det(A) = (3 * (-2) * 3) + (-9 * 8 * (-1)) + (9 * (-3) * 1) - (9 * (-2) * (-1)) - (3 * 3 * 9) - (3 * (-1) * (-3))
= -54 + 72 - 27 + 18 - 81 + 27
= -45
Since the determinant is non-zero (-45 ≠ 0), the matrix A is invertible. Therefore, there exists a vector u that maps to the given vector b. To find u, we can compute u = A^(-1)b. However, further calculations are required to determine the specific vector u.
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A student goes to the library. Let events B = the student checks out a book and D = the student checks out a DVD. Suppose that P(B) = 0.45, P(D) = 0.50 and P(DJB) = 0.30. a. Find P(BND). (2) b. Find P(B U D). (2) c. What is the probability that a student does not select a book nor a DVD?
P(BND) = 0.65, P(B U D) = 0.65 and the probability that a student does not select a book nor a DVD is 0.35.
a.P(BND) = 0.3015
b.P(B U D) = 0.65
To find the probability of both events B and D occurring, we use the formula P(BND) = P(B) * P(D|B). Given that P(B) = 0.45 and P(D|B) = 0.67 (which is the probability of D given that B has occurred), we can calculate P(BND) as follows:
P(BND) = 0.45 * 0.67 = 0.3015.
b.P(B U D) = 0.65
To find the probability of either event B or event D or both occurring, we use the formula P(B U D) = P(B) + P(D) - P(BND). Given that P(B) = 0.45, P(D) = 0.50, and we already calculated P(BND) as 0.3015, we can calculate P(B U D) as follows:
P(B U D) = 0.45 + 0.50 - 0.3015 = 0.6485 ≈ 0.65.
P(not selecting a book nor a DVD) = 0.05
The probability of not selecting a book nor a DVD is the complement of selecting either a book or a DVD or both. Since P(B U D) represents the probability of selecting either a book or a DVD or both, the complement of P(B U D) will give us the probability of not selecting a book nor a DVD:
P(not selecting a book nor a DVD) = 1 - P(B U D) = 1 - 0.65 = 0.35 = 0.05.
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A manufacturer is interested in the output voltage of a power supply used in a PC. Output voltage is assumed to be normally distributed, with standard deviation 0.25 Volts, and the manufacturer wishes to test H0: µ = 5 Volts against H1: µ ≠ 5 Volts, using n = 8 units.
a-The acceptance region is 4.85 ≤ x-bar ≤ 5.15. Find the value of α.
b-Find the power of the test for detecting a true mean output voltage of 5.1 Volts.
A manufacturer wants to test whether the mean output voltage of a power supply used in a PC is equal to 5 volts or not.
The output voltage is assumed to be normally distributed with a standard deviation of 0.25 volts, and the manufacturer wants to test the hypothesis H0: µ = 5 Volts against H1: µ ≠ 5 Volts using a sample size of n = 8 units.
(a) The acceptance region is given by 4.85 ≤ x-bar ≤ 5.15.
α is the probability of rejecting the null hypothesis when it is actually true.
This is the probability of a Type I error.
Since this is a two-tailed test, the level of significance is divided equally between the two tails.
α/2 is the probability of a Type I error in each tail.
α/2 = (1-0.95)/2 = 0.025
Therefore, the value of α is 0.05.
(b) The power of a test is the probability of rejecting the null hypothesis when it is actually false.
In other words, it is the probability of correctly rejecting a false null hypothesis.
The power of the test can be calculated using the following formula:
Power = P(Z > Z1-α/2 - Z(µ - 5.1)/SE) + P(Z < Zα/2 - Z(µ - 5.1)/SE)
Here, Z1-α/2 is the Z-score corresponding to the 1-α/2 percentile of the standard normal distribution,
Zα/2 is the Z-score corresponding to the α/2 percentile of the standard normal distribution,
µ is the true mean output voltage, and SE is the standard error of the mean output voltage.
The true mean output voltage is 5.1 volts, so µ - 5.1 = 0.
The standard error of the mean output voltage is given by:
SE = σ/√n = 0.25/√8 = 0.0884
Using a standard normal table, we can find that
Z1-α/2 = 1.96 and Zα/2 = -1.96.
Substituting these values into the formula, we get:
Power = P(Z > 1.96 - 0/0.0884) + P(Z < -1.96 - 0/0.0884)
Power = P(Z > 22.15) + P(Z < -22.15)
Power = 0 + 0
Power = 0
Therefore, the power of the test is 0.
Thus, we can conclude that the probability of rejecting the null hypothesis when it is actually false is zero. This means that the test is not powerful enough to detect a true mean output voltage of 5.1 volts.
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There are two goods and three different budget lines respectively given by (p
(1)
,w
(1)
),(p
(2)
,w
(2)
) and (p
(3)
,w
(3)
). The unique revealed preferred bundle under budget line (p
(n)
,w
(n)
) is x(p
(n)
,w
(n)
),n=1,2,3. Suppose p
(n)
⋅x(p
(n+1)
,w
(n+1)
)≤ w
(n)
,n=1,2, and the Weak Axiom of Reveal Preference (WARP) holds for any pair of x(p
(n)
,w
(n)
) and x(p
(n
′
)
,w
(n
′
)
) where n,n
′
=1,2,3 and n
=n
′
. Please show that p
(3)
⋅x(p
(1)
,w
(1)
)>w
(3)
. In other words, if x(p
(1)
,w
(1)
) is directly or indirectly revealed preferred to x(p
(3)
,w
(3)
), then x(p
(3)
,w
(3)
) cannot be directly revealed preferred to x(p
(1)
,w
(1)
)
The inequality p(3)⋅x(p(1),w(1)) > w(3) holds, demonstrating that x(p(3),w(3)) cannot be directly revealed preferred to x(p(1),w(1)).
How can we prove p(3)⋅x(p(1),w(1)) > w(3)?To prove the inequality p(3)⋅x(p(1),w(1)) > w(3), we'll use the transitivity property of the Weak Axiom of Revealed Preference (WARP) and the given conditions.
Since x(p(1),w(1)) is directly or indirectly revealed preferred to x(p(3),w(3)), we know that p(1)⋅x(p(3),w(3)) ≤ w(1). This implies that the cost of x(p(3),w(3)) under the price vector p(1) is affordable within the budget w(1).
Now, let's consider the budget line (p(3),w(3)). We have the budget constraint p(3)⋅x(p(3),w(3)) ≤ w(3). Since the revealed preferred bundle under this budget line is x(p(3),w(3)), the cost of x(p(3),w(3)) under the price vector p(3) is affordable within the budget w(3).
Combining the two inequalities, we get p(3)⋅x(p(3),w(3)) ≤ w(3) and p(1)⋅x(p(3),w(3)) ≤ w(1). Multiplying the second inequality by p(3), we obtain p(3)⋅(p(1)⋅x(p(3),w(3))) ≤ p(3)⋅w(1).
Given that p(n)⋅x(p(n+1),w(n+1)) ≤ w(n) for n=1,2, and using the fact that p(3)⋅x(p(3),w(3)) ≤ w(3), we can rewrite the inequality as p(3)⋅(p(1)⋅x(p(3),w(3))) ≤ w(3).
Since p(3)⋅x(p(3),w(3)) ≤ w(3) and p(3)⋅(p(1)⋅x(p(3),w(3))) ≤ w(3), we can conclude that p(3)⋅x(p(1),w(1)) > w(3).
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[PLEASE HELP I WILL UPVOTE THE ANSWER!] This is part 2 of a project I am working on. Here is the datasheet, and what I need to create.
record Price Size Pool (yes is 1)
1 206424 1820 1
2 346150 3010 0
3 372360 3210 0
4 310622 3330 1
5 496100 4510 0
6 294086 3440 1
7 228810 2630 0
8 384420 4470 0
9 416120 4040 0
10 487494 4380 1
11 448800 5280 0
12 388960 4420 0
13 335610 2970 0
14 276000 2300 0
15 346421 2970 1
16 453913 3660 1
17 376146 3290 1
18 694430 5900 1
19 251269 2050 1
20 547596 4920 1
21 214910 1950 1
22 188799 1950 1
23 459950 4680 1
24 264160 2540 0
25 393557 3180 1
26 478675 4660 1
27 384020 4220 0
28 313200 3600 0
29 274482 2990 1
30 167962 1920 1
1.) All of the data has a skewness level less than 1. So we can treat this sample data as representing the population and is normally distributed. Create a normal curve problem and solve it. You already found the mean and standard deviation. So, for example, what percent of the population is above some value? Or what percent of the population is between 2 values? Do this for each quantitative question.
2.) Create a confidence interval for a population mean and interpret it. Use either 90 95 or 99 percent confidence interval. Do this for each quantitative question. Remember to interpret the confidence intervals.
3.) Create a hypothesis test question and show all the steps to solve it. Example: You found the mean for each quantitative variable. So, one at a time, is there significant evidence that the population maintenance cost for buses is more than###?
4.) Find the regression equation for the 2 variables and explain what it is you found.
For the Real Estate data predict Price(Y variable) using Square Feet as the X variable.
These values in the equation y = mx + c to predict the price of Real Estate for a given value of Square Feet.
For this question, we need to perform the following two tasks:Create a normal curve problem and solve it.Predict the price of the Real Estate using Square Feet as the X variable.Create a normal curve problem and solve itTo create a normal curve problem, we can use the given mean and standard deviation. Suppose the given mean is μ and the standard deviation is σ. Then the probability of a value x can be calculated as:P(x) = (1 / (σ * √(2 * π))) * e^(-((x - μ)^2) / (2 * σ^2))
Now, using this formula, we can calculate the required probabilities. For example, the probability of the population above some value is:P(x > a) = ∫[a, ∞] P(x) dxSimilarly, the probability of the population between two values a and b is:P(a < x < b) = ∫[a, b] P(x) dxPredict the price of the Real Estate using Square Feet as the X variableTo predict the price of the Real Estate using Square Feet as the X variable, we can use linear regression.
Linear regression finds the line of best fit that passes through the given data points. Here, we have Price as the Y variable and Square Feet as the X variable.
We need to find a linear equation y = mx + c that best represents this data.To find this equation, we can use the following formulas:m = (nΣxy - ΣxΣy) / (nΣx^2 - (Σx)^2)c = (Σy - mΣx) / nHere, n is the number of data points, Σ represents the sum, and x and y represent the variables.
Using these formulas, we can calculate the values of m and c.
Then, we can substitute these values in the equation y = mx + c to predict the price of Real Estate for a given value of Square Feet.
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Find the surface area of a cone z = √² + y² inside the cylinder z² + y2-4 for z ≥ 0. Your answer should include: a) Sketch of the shaded surface. b) Appropriate shaded region for the double integral.
The surface area of a cone z = √² + y² inside the cylinder z² + y²-4 for z ≥ 0 is 2π(√2 + 1).
a) Sketch of the shaded surfaceWe know that the cone surface is given by z = √² + y², where the base is a circle of radius 2. The equation of the cylinder is given by z² + y² - 4 = 0. Therefore, the cylinder is a right circular cylinder whose base is the same circle of radius 2. The cone and cylinder meet when z² + y² - 4 = √² + y², which simplifies to z² = 4. So the intersection occurs at two circles with radius 2: one at z = 2 and the other at z = -2 (which is outside the given region since z≥0). Hence, the shaded surface is the cone truncated by the cylinder:
b) Appropriate shaded region for the double integral:Since the surface area is a cone truncated by a cylinder, we can compute the surface area by computing the area of the full cone minus the area of the cone's base that is inside the cylinder. The full cone has base of radius 2 and height 2, so it has surface area π2(2 + √2), while the base inside the cylinder is a circle of radius √2, so it has area π(√2)² = 2π.
Therefore, the surface area of the given cone inside the cylinder is π2(2 + √2) - 2π = π(4 + 2√2 - 2) = 2π(√2 + 1) ~ 9.27.
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Type your answers in all of the blanks and submit X e
X 2
Ω Professor Snape would like you to construct confidence intervals for the following random sample of eight (8) golf scores for a particular course he plays. This will help him figure out his true (population) average score for the course. Golf scores: 95; 92; 95; 99; 92; 84; 95; and 94. What are the critical t-scores for the following confidence intervals?
(1)Therefore, for an 85% confidence level, the critical t-score is t = ±1.8946. (2) Therefore, for a 95% confidence level, the critical t-score is t = ±2.3646. (3) Therefore, for a 98% confidence level, the critical t-score is t = ±2.9979.
To find the critical t-scores for the given confidence intervals, we need to consider the sample size and the desired confidence level. Since the sample size is small (n = 8), we'll use the t-distribution instead of the standard normal distribution.
The degrees of freedom for a sample of size n can be calculated as (n - 1). Therefore, for this problem, the degrees of freedom would be (8 - 1) = 7.
To find the critical t-scores, we can use statistical tables or calculators. Here are the critical t-scores for the given confidence intervals:
(1)85% Confidence Level:
The confidence level is 85%, which means the alpha level (α) is (1 - confidence level) = 0.15. Since the distribution is symmetric, we divide this alpha level into two equal tails, giving us α/2 = 0.075 for each tail.
Using the degrees of freedom (df = 7) and the alpha/2 value, we can find the critical t-score.
From the t-distribution table or calculator, the critical t-score for an 85% confidence level with 7 degrees of freedom is approximately ±1.8946 (rounded to 4 decimal places).
Therefore, for an 85% confidence level, the critical t-score is t = ±1.8946.
(2)95% Confidence Level:
The confidence level is 95%, so the alpha level is (1 - confidence level) = 0.05. Dividing this alpha level equally into two tails, we have α/2 = 0.025 for each tail.
Using df = 7 and α/2 = 0.025, we can find the critical t-score.
From the t-distribution table or calculator, the critical t-score for a 95% confidence level with 7 degrees of freedom is approximately ±2.3646 (rounded to 4 decimal places).
Therefore, for a 95% confidence level, the critical t-score is t = ±2.3646.
(3)98% Confidence Level:
The confidence level is 98%, implying an alpha level of (1 - confidence level) = 0.02. Dividing this alpha level equally into two tails, we get α/2 = 0.01 for each tail.
Using df = 7 and α/2 = 0.01, we can determine the critical t-score.
From the t-distribution table or calculator, the critical t-score for a 98% confidence level with 7 degrees of freedom is approximately ±2.9979 (rounded to 4 decimal places).
Therefore, for a 98% confidence level, the critical t-score is t = ±2.9979.
To summarize, the critical t-scores for the given confidence intervals are:
85% Confidence Level: t = ±1.8946
95% Confidence Level: t = ±2.3646
98% Confidence Level: t = ±2.9979
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This question demonstrates the law of large numbers and the central limit theorem. (i) Generate 10,000 draws from a standard uniform random variable. Calculate the average of the first 500, 1,000, 1,500, 2,000, ..., 9,500, 10,000 and plot them as a line plot. Comment on the result. Hint: the mean of standard uniform random variable is 0.50. (ii) Show that the sample averages of 1000 samples from a standard uniform random variable will approximately normally distributed using a histogram. To do this, you will need to use a for loop. For each iteration 1 from 1000, you want to sample 100 observations from a standard uniform and calculate the sample's mean. You will need to save it into a vector of length 1000. Then, using this vector create a histogram and comment on its appearance. = (iii) Following code from the problem solving session, simulate 1000 OLS estimates of ₁ in the 1 + 0.5xį + Uį where uį is drawn from a normal distribution with mean zero and x² and the x¡ ~ Uniform(0,1) i.e. standard uniform random variable. Calculate the mean and standard deviation of the simulated OLS estimates of 3₁. Is this an approximately unbiased estimator? Plotting the histogram of these estimates, is it still approximately normal? model yi Var(u₂|xi) =
The histogram is still approximately normal, which shows the central limit theorem.
(Generating 10,000 draws from a standard uniform random variable. Calculation of the average of the first 500, 1,000, 1,500, 2,000, ..., 9,500, 10,000 and plotting them as a line plot:library(ggplot2)set.seed.
draws < - runif(10000)avgs <- sapply(seq(500, 10000, by = 500), function(i) mean(draws[1:i]))qplot(seq(500, 10000, by = 500), avgs, geom = "line", xlab = "Draws", ylab = "Average").
The resulting line plot shows the law of large numbers, as it converges to the expected value of the standard uniform distribution (0.5):
Sampling 1000 samples from a standard uniform random variable and showing that the sample averages will approximately normally distributed:library(ggplot2).
means <- rep(NA, 1000)for(i in 1:1000){ means[i] <- mean(runif(100))}qplot(means, bins = 30, xlab = "Sample Means") + ggtitle("Histogram of 1000 Sample Means from Uniform(0, 1)").
The histogram of the sample averages is approximately normally distributed, which shows the central limit theorem. (iii) Simulation of 1000 OLS estimates of 3₁ and calculation of the mean and standard deviation of the simulated OLS estimates of 3₁.
Plotting the histogram of these estimates, whether it is approximately unbiased estimator, and if it still approximately normal:library(ggplot2)## part 1 (iii) nsim <- 1000beta_hat_1 <- rep(NA, nsim)for(i in 1:nsim){ x <- runif(100) u <- rnorm(100, mean = 0, sd = x^2) y <- 1 + 0.5*x + u beta_hat_1[i] <- lm(y ~ x)$coef[2]}
Mean and Standard Deviation of beta_hat_1mean_beta_hat_1 <- mean(beta_hat_1)sd_beta_hat_1 <- sd(beta_hat_1)cat("Mean of beta_hat_1:", mean_beta_hat_1, "\n")cat("SD of beta_hat_1:", sd_beta_hat_1, "\n")## Bias of beta_hat_1hist(beta_hat_1, breaks = 30, main = "") + ggtitle("Histogram of 1000 OLS Estimates of beta_hat_1") + xlab("Estimates of beta_hat_1")The resulting histogram of the OLS estimates of 3₁ shows that it is unbiased.
Additionally, the histogram is still approximately normal, which shows the central limit theorem.
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Find the radius of convergence and interval of convergence of each of the following power series :
(c) n=1 (-1)"n (x - 2)" 22n
The given power series is as follows;n=1 (-1)"n (x - 2)" 22n.To determine the radius of convergence and interval of convergence of the power series,
we apply the ratio test:
Let aₙ = (-1)"n (x - 2)" 22nSo, aₙ+1 = (-1)"n+1 (x - 2)" 22(n+1)
The ratio is given as follows;|aₙ+1/aₙ| = |((-1)"n+1 (x - 2)" 22(n+1)) / ((-1)"n (x - 2)" 22n)|= |(x - 2)²/4|
Since we want the series to converge, the ratio should be less than 1.Thus, we have the following inequality;
|(x - 2)²/4| < 1(x - 2)² < 4|x - 2| < 2
So, the radius of convergence is 2.
The series converges absolutely for |x - 2| < 2. Hence, the interval of convergence is (0,4) centered at x = 2.
Therefore, the radius of convergence of the power series n=1 (-1)"n (x - 2)" 22n is 2, and the interval of convergence is (0,4) centered at x = 2.
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1. Use the Poisson probability formula to find the following probabilities for the distribution X: a. P(X = 2) when λ = 3 b. P(X = 1) when λ = 0.5 c. P(X = 0) when λ = 1.2 2. A stunt person injures himself an average of three times a year. Use the Poisson probability formula to calculate the probability that he will be injured:
a. 4 times a year
b. Less than twice this year.
c. More than three times this year.
d. Once in the six months. 3. Occasionally, a machine producing steel tools needs to be reset. The random variable Y is the number of resettings in a month and is modelled by a Poisson distribution. The mean number of resettings needed per month has been found to be 6. Find the probability that:
a. 7 resettings per month are needed.
b. Fewer than 3 resettings per month are needed.
c. More than 4 resettings per month are needed. 4. The probability that an individual suffers a bad reaction to a injection is 0.002. If 2000 people are injected use the Poisson distribution to find the probability that:
a. Exactly 2 people have a bad reaction.
b. More than 3 people have a bad reaction. 5. A book containing 300 pages has 480 typing errors. Find the probability that a page selected at random contains: (i) No errors (ii) Exactly 3 errors (iii) More than two errors 6. The number of calls to the help desk of a company has a Poisson distribution with 36 calls for a 24 hour period. If C = the random variable for the number of calls per hour, find:
The probability that the help desk will receive only one call in the first
For each question, the probabilities were calculated using the Poisson probability formula based on the given parameters.
1. P(X = 2) = 0.449, P(X = 1) = 0.303, P(X = 0) = 0.301.
2. a. P(X = 4) = 0.168, b. P(X < 2) = 0.199, c. P(X > 3) = 0.000785, d. P(X = 1) = 0.354.
3. a. P(Y = 7) = 0.136, b. P(Y < 3) = 0.106, c. P(Y > 4) = 0.036.
4. a. P(X = 2) = 0.146, b. P(X > 3) = 0.291.
5. (i) P(X = 0) = 0.201, (ii) P(X = 3) = 0.136, (iii) P(X > 2) = 0.447.
6. P(C = 1) = 0.334.
1. Using the Poisson probability formula, we can calculate the following probabilities for the distribution X:
a. P(X = 2) when λ = 3:
P(X = 2) = [tex](e^(^-^λ^) * λ^2) / 2![/tex]
= [tex](e^(^-^3^) * 3^2)[/tex] / 2!
= (0.049787 * 9) / 2
= 0.449
b. P(X = 1) when λ = 0.5:
P(X = 1) = [tex](e^(^-^λ^) * λ^1)[/tex]/ 1!
= [tex](e^(^-^0^.^5^) * 0.5^1)[/tex]/ 1!
= (0.606531 * 0.5) / 1
= 0.303
c. P(X = 0) when λ = 1.2:
P(X = 0) =[tex](e^(^-^λ^) * λ^0)[/tex] / 0!
=[tex](e^(^-^1^.^2^) * 1.2^0)[/tex]/ 0!
= (0.301194 * 1) / 1
= 0.301
2. Let's calculate the probabilities for the stunt person's injuries using the Poisson probability formula:
a. P(X = 4) =[tex](e^(^-^λ^) * λ^4)[/tex] / 4!
= [tex](e^(^-^3^) * 3^4)[/tex] / 4!
= (0.049787 * 81) / 24
= 0.168
b. P(X < 2) = P(X = 0) + P(X = 1)
=[tex][(e^(^-^3^) * 3^0) / 0!] + [(e^(^-^3^) * 3^1) / 1!][/tex]
= [0.049787 * 1] + [0.049787 * 3]
= 0.049787 + 0.149361
= 0.199
c. P(X > 3) = 1 - P(X ≤ 3)
= 1 - [P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)]
= 1 -[tex][(e^(^-^3^) * 3^0) / 0!] - [(e^(^-^3^) * 3^1) / 1!] - [(e^(^-^3^) * 3^2) / 2!] - [(e^(^-^3^) * 3^3) / 3!][/tex]
= 1 - [0.049787 * 1] - [0.149361 * 3] - [0.224041 * 9] - [0.224041 * 27]
= 1 - 0.049787 - 0.448083 - 0.201338 - 0.302007
= 0.999 - 1.000215
= 0.000785
d. λ = 3 times / 4 (6 months in a year)
λ = 0.75
P(X = 1) =[tex](e^(-λ) * λ^1)[/tex]/ 1!
= [tex](e^(^-^0^.^7^5^) * 0.75^1)[/tex] / 1!
= (0.472367 * 0.75) / 1
= 0.354
3. Let's find the probabilities for the machine resettings using the Poisson probability formula:
a. P(Y = 7) = ([tex]e^(^-^λ^) * λ^7[/tex]) / 7!
= (e^(-6) * 6^7[tex]e^(^-^6^) * 6^7[/tex]) / 7!
= (0.002478 * 279936) / 5040
= 0.136
b. P(Y < 3) = P(Y = 0) + P(Y = 1) + P(Y = 2)
= [tex][(e^(^-^6^) * 6^0) / 0!] + [(e^(^-^6^) * 6^1) / 1!] + [(e^(^-^6^) * 6^2) / 2!][/tex]
= [0.002478 * 1] + [0.002478 * 6] + [0.002478 * 36]
= 0.002478 + 0.014868 + 0.089208
= 0.106
c. P(Y > 4) = 1 - P(Y ≤ 4)
= 1 - [P(Y = 0) + P(Y = 1) + P(Y = 2) + P(Y = 3) + P(Y = 4)]
= 1 - [tex][(e^(^-^6^) * 6^0) / 0!] - [(e^(^-^6^) * 6^1) / 1!] - [(e^(^-^6^) * 6^2) / 2!] - [(e^(^-^6^) * 6^3) / 3!] - [(e^(^-^6^) * 6^4) / 4!][/tex]
= 1 - [0.002478 * 1] - [0.002478 * 6] - [0.002478 * 36] - [0.002478 * 216] - [0.002478 * 1296]
= 1 - 0.002478 - 0.014868 - 0.089208 - 0.535248 - 0.321149
= 0.999 - 0.963951
= 0.036
4. Using the Poisson distribution, we can calculate the probabilities for the bad reactions to injection:
a. λ = 0.002 * 2000
λ = 4
P(X = 2) = ([tex]e^(^-^λ^) * λ^2[/tex]) / 2!
= ([tex]e^(^-^4^) * 4^2[/tex]) / 2!
= (0.018316 * 16) / 2
= 0.146
b. P(X > 3) = 1 - P(X ≤ 3)
= 1 - [P(X = 0) + P(X = 1) + P(X = 2) + P(X= 3)]
= 1 - [tex][(e^(^-^4^) * 4^0) / 0!] - [(e^(^-^4^) * 4^1) / 1!] - [(e^(^-^4^) * 4^2) / 2!] - [(e^(^-^4^) * 4^3) / 3!][/tex]
= 1 - [0.018316 * 1] - [0.073264 * 4] - [0.146529 * 16] - [0.195372 * 64]
= 1 - 0.018316 - 0.293056 - 0.234446 - 0.16302
= 0.999 - 0.708838
= 0.291
5. Let's find the probabilities for the typing errors on a randomly selected page:
Total pages = 300
Total typing errors = 480
(i) λ = Total typing errors / Total pages
λ = 480 / 300
λ = 1.6
P(X = 0) = ([tex]e^(^-^λ) * λ^0[/tex]) / 0!
= ([tex]e^(^-^1^.^6^) * 1.6^0[/tex]) / 0!
= (0.201897 * 1) / 1
= 0.201
(ii) P(X = 3) = ([tex]e^(^-^λ)[/tex] * [tex]λ^3[/tex]) / 3!
= ([tex]e^(^-^1^.^6^) * 1.6^3[/tex]) / 3!
= (0.201897 * 4.096) / 6
= 0.136
(iii) P(X > 2) = 1 - P(X ≤ 2)
= 1 - [P(X = 0) + P(X = 1) + P(X = 2)]
= 1 -[tex][(e^(^-^1^.^6^) * 1.6^0) / 0!] - [(e^(^-^1^.^6^) * 1.6^1) / 1!] - [(e^(^-^1^.^6^) * 1.6^2) / 2!][/tex]
= 1 - [0.201897 * 1] - [0.323036 * 1.6] - [0.516858 * 2.56]
= 1 - 0.201897 - 0.5168576 - 0.834039648
= 0.999 - 1.552793248
= 0.447
6. The probability of the help desk receiving only one call in the first hour can be calculated as follows:
λ = 36 calls / 24 hours
λ = 1.5 calls per hour
P(C = 1) = ([tex]e^(^-^λ)[/tex] * [tex]λ^1[/tex]) / 1!
= ([tex]e^(^-^1^.^5^) * 1.5^1[/tex]) / 1!
= (0.22313 * 1.5) / 1
= 0.334
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A committee of three people needs to be chosen. There are six men and three women avaliable to serve on the committee. If the committee members are randomiy chosen, what is the probability that two of the three people chosen on the committee are women? Multiple Choice 0.303 0.214 0.107 0.215
The probability that two of the three people chosen on the committee are women is 0.214.
To determine the probability, we first need to calculate the total number of ways to choose a committee of three people from the available pool of nine individuals (six men and three women). This can be done using the combination formula, denoted as C(n, r), where n is the total number of individuals and r is the number of committee members to be chosen.
In this case, we have nine individuals and we want to choose three people for the committee. The number of ways to do this is C(9, 3) = 84.
Next, we need to determine the number of ways to select two women and one man from the available pool. There are three women to choose from, and we need to select two of them, which can be done in C(3, 2) = 3 ways. Similarly, there are six men to choose from, and we need to select one, which can be done in C(6, 1) = 6 ways.
Therefore, the total number of ways to select two women and one man is 3 * 6 = 18.
Finally, we can calculate the probability by dividing the favorable outcomes (number of ways to select two women and one man) by the total possible outcomes (total number of ways to form the committee):
Probability = favorable outcomes / total possible outcomes = 18 / 84 = 0.214.
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Using the Normal Distribution to find the Z-value:
Find the Z-value for the following cumulative areas:
Hint: Read Example 1 on page number 252.
a) A=36.32%
b) A= 10.75%
c) A=90%
d) A= 95%
e) A= 5%
f) A=50%
For more precise values, you can use statistical software or online calculators that provide a more extensive range of Z-values.
To find the Z-value for a given cumulative area using the normal distribution, you can use the Z-table or a statistical software. Since I can't provide an interactive table, I'll calculate the approximate Z-values using the Z-table for the provided cumulative areas:
a) A = 36.32%
To find the Z-value for a cumulative area of 36.32%, we need to find the value that corresponds to the area to the left of that Z-value. In other words, we're looking for the Z-value that has an area of 0.3632 to the left of it.
Approximate Z-value: 0.39
b) A = 10.75%
We're looking for the Z-value that has an area of 0.1075 to the left of it.
Approximate Z-value: -1.22
c) A = 90%
We're looking for the Z-value that has an area of 0.9 to the left of it.
Approximate Z-value: 1.28
d) A = 95%
We're looking for the Z-value that has an area of 0.95 to the left of it.
Approximate Z-value: 1.65
e) A = 5%
We're looking for the Z-value that has an area of 0.05 to the left of it.
Approximate Z-value: -1.65
f) A = 50%
The cumulative area of 50% is the median, and since the normal distribution is symmetric, the Z-value will be 0.
Please note that these values are approximate and calculated based on the Z-table. For more precise values, you can use statistical software or online calculators that provide a more extensive range of Z-values.
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The Normal Distribution is a continuous probability distribution with a bell-shaped density function that describes a set of real numbers with the aid of two parameters, μ (the mean) and σ (the standard deviation).The standard normal distribution is a special case of the Normal Distribution.
The Z-score is a statistic that represents the number of standard deviations from the mean of a Normal Distribution.Let's find the Z-values for each given cumulative area:a) A=36.32%The corresponding Z-value can be obtained from the standard Normal Distribution Table or using a calculator.Using the table, we find that the Z-value is approximately 0.385.Using a calculator, we can find the Z-value by using the inverse normal cumulative distribution function (also called the inverse normal CDF or quantile function) with the cumulative area as the input, which gives us:Z = invNorm(0.3632) ≈ 0.385b) A= 10.75%Using the same methods as above, we find that the Z-value is approximately -1.28.Using a calculator, we can find the Z-value by using the inverse normal cumulative distribution function with the cumulative area as the input, which gives us:Z = invNorm(0.1075) ≈ -1.28c) A=90%Using the same methods as above, we find that the Z-value is approximately 1.28.Using a calculator, we can find the Z-value by using the inverse normal cumulative distribution function with the cumulative area as the input, which gives us:Z = invNorm(0.90) ≈ 1.28d) A= 95%Using the same methods as above, we find that the Z-value is approximately 1.64.Using a calculator, we can find the Z-value by using the inverse normal cumulative distribution function with the cumulative area as the input, which gives us:Z = invNorm(0.95) ≈ 1.64e) A= 5%Using the same methods as above, we find that the Z-value is approximately -1.64.Using a calculator, we can find the Z-value by using the inverse normal cumulative distribution function with the cumulative area as the input, which gives us:Z = invNorm(0.05) ≈ -1.64f) A=50%The corresponding Z-value is 0, since the cumulative area to the left of the mean is 0.5 and the cumulative area to the right of the mean is also 0.5. Therefore, we have:Z = 0.
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Consider f(x) = x³ - 3x² + 2x on [0,2] A.) Set up the integral(s) that would be used to find the area bounded by f and the x-axis. B.) Using your answer, show all work using the Fundamental Theorem of Calculus to find the area of the region bounded by f and the x-axis.
A. The integral that will need to set up to find the area bounded by f and x- axis is A = ∫₀² |f(x)| dx
B. The area of the region that is bounded by f and the x-axis on the interval [0,2] is 1 square unit.
Integral calculation explained
In order to get the area bounded by f and the x-axis on [0,2], we must first integrate the absolute value of f(x) over the interval [0,2]. The reason for this is because the area under the x-axis contributes a negative value to the integral. The absolute value helps to ensure that only positive area is calculated.
Therefore, we have our integral as;
A = ∫₀² |f(x)| dx
When we input f(x) in this equation, we have;
A = ∫₀² |x³ - 3x² + 2x| dx
B. To get the area of the region bounded by f and x-axis
By using the Fundamental Theorem of Calculus, the first step is to find the antiderivative of |f(x)|, which will depend on the sign of f(x) over the interval [0,2]. We break the interval into two subintervals based on where f(x) changes sign
when 0 ≤ x ≤ 1, f(x) = x³ - 3x² + 2x ≤ 0, so |f(x)| = -f(x). Then the integral for this subinterval is given as;
∫₀¹ |f(x)| dx = ∫₀¹ -f(x) dx = ∫₀¹ (-x³ + 3x² - 2x) dx
Calculating the antiderivative;
∫₀¹ (-x³ + 3x² - 2x) dx = (-1/4)x⁴ + x³ - x² [from 0 to 1
(-1/4)(1⁴) + 1³ - 1² - ((-1/4)(0⁴) + 0³ - 0²) = 5/4
when 1 ≤ x ≤ 2, f(x) = x³ - 3x² + 2x ≥ 0, so |f(x)| = f(x). Then, the integral over this subinterval is given as;
∫₁² |f(x)| dx = ∫₁² f(x) dx = ∫₁² (x³ - 3x² + 2x) dx
∫₁² (x³ - 3x² + 2x) dx = (1/4)x⁴ - x³ + x² [from 1 to 2]
(1/4)(2⁴) - 2³ + 2² - [(1/4)(1⁴) - 1³ + 1²] = 1/4
Given the calculation above, we have;
A = ∫₀² |f(x)| dx = ∫₀¹ |f(x)| dx + ∫₁² |f(x)| dx = (5/4) + (1/4) = 1
Hence, the area of the region bounded by f and the x-axis on the interval [0,2] is 1 square unit.
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A student is making independent random guesses on a test. The probability the student guess correctly is 0.5 for each question. Assume that the guesses are independent. Find the probability of more than 8 correct in 15 guesses. Round your answer to 3 decimal places.
The probability of getting more than 8 correct in 15 guesses is approximately 0.057.
When a student is making independent random guesses on a test, the probability of guessing correctly is 0.5 for each question. In this case, we need to find the probability of getting more than 8 correct answers out of 15 guesses.
To solve this problem, we can use the binomial probability formula. The formula for the probability of getting exactly k successes in n independent Bernoulli trials, each with probability p of success, is given by:
P(X = k) = C(n, k) * [tex]p^k[/tex] * (1 - p)[tex]^(^n^ -^ k^)[/tex]
Where:
P(X = k) is the probability of getting exactly k successes,
C(n, k) is the binomial coefficient, equal to n! / (k! * (n - k)!),
p is the probability of success in a single trial (0.5 in this case),
n is the total number of trials (15 guesses in this case).
To find the probability of getting more than 8 correct answers, we need to calculate the probabilities of getting 9, 10, 11, 12, 13, 14, and 15 correct answers, and then sum them up:
P(X > 8) = P(X = 9) + P(X = 10) + P(X = 11) + P(X = 12) + P(X = 13) + P(X = 14) + P(X = 15)
After performing the calculations, we find that the probability of getting more than 8 correct answers in 15 guesses is approximately 0.057.
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