(a) The center of the sphere is in the first octant and is tangent to the xz-plane and to the yz-plane. This means that the center of the sphere is at a point of the form (a,b,5) where a,b≥0. The distance from the origin to the center of the sphere is [tex]\sqrt{43}[/tex], so we have [tex]x^{2} +x^{2} +(5-0)^{2} =43[/tex] This gives us [tex]a^{2} +b^{2} =38[/tex]
The radius of the sphere is the distance from the center of the sphere to the point where the sphere touches the xz-plane. This distance is equal to the length of the hypotenuse of a right triangle with legs of length a and b. Therefore, the radius of the sphere is [tex]\sqrt{a^{2}+ b^{2} } =\sqrt{38}[/tex]
The equation of the sphere is [tex](x-a)^{2}+ (y-b)^{2}+ (z-5)^{2} =38[/tex]
(b) The point where the sphere touches the xz-plane is (a,0,5). The distance between the origin and this point is [tex]\sqrt{a} ^{2}+\sqrt(5-0)^{2} =\sqrt{a^{2} +25}[/tex]
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[6] College presidents receive a housing provision with an annual mean of $50,000. Assume that a normal distribution applies and that the standard deviation is $5,000. A. What percentage of college presidents receive an annual housing provision exceeding $45,000 per year? B. What percentage of college presidents receive an annual housing provision between $39,500 and $47,200 per year? C. Find the housing provision such that 17.36% of college presidents receive an amount exceeding this figure.
(a) To find the percentage of college presidents receiving an annual housing provision exceeding $45,000 per year, we need to calculate the probability of a value greater than $45,000 based on the given normal distribution with a mean of $50,000 and a standard deviation of $5,000.
(b) To find the percentage of college presidents receiving an annual housing provision between $39,500 and $47,200 per year, we calculate the probability of a value falling within this range based on the normal distribution.
(c) To determine the housing provision such that 17.36% of college presidents receive an amount exceeding this figure, we find the corresponding value of the housing provision using the cumulative distribution function (CDF) of the normal distribution.
(a) Using the normal distribution, we can calculate the probability of a value exceeding $45,000 by finding the area under the curve to the right of $45,000. This can be done by standardizing the value using the formula z = (x - μ) / σ, where x is the value ($45,000), μ is the mean ($50,000), and σ is the standard deviation ($5,000). Then, we can look up the corresponding z-score in the standard normal distribution table to find the probability.
(b) To calculate the percentage of college presidents receiving an annual housing provision between $39,500 and $47,200 per year, we need to find the probabilities of values falling below $47,200 and $39,500 separately and then subtract the two probabilities. Similar to (a), we standardize the values and use the standard normal distribution table to find the probabilities.
(c) To find the housing provision such that 17.36% of college presidents receive an amount exceeding this figure, we need to find the value that corresponds to the 17.36th percentile of the normal distribution. This can be done by finding the z-score that corresponds to the desired percentile using the standard normal distribution table, and then converting it back to the original scale using the formula x = μ + zσ, where x is the desired value, μ is the mean, z is the z-score, and σ is the standard deviation.
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You are given the point (3, 0) in polar coordinates. (i) Find another pair of polar coordinates for this point such that r> 0 and 2 ≤ 0 <4TT. ↑ = sqrt9 0 = (ii) Find another pair of polar coordinates for this point such that r < 0 and 0 ≤ 0 < 2TT. ↑ = 0 = (b) You are given the point (2,-/7) in polar coordinates. Find another pair of polar coordinates for this point such that r> 0 and 2π ≤ 0 <4TT. r = 0 = (ii) Find another pair of polar coordinates for this point such that r < 0 and -2π ≤ 0 <0. P = 0 = (c) You are given the point (-1, -T/2) in polar coordinates. (i) Find another pair of polar coordinates for this point such that r> 0 and 2π ≤ 0 <4TT. r = 0 = (ii) Find another pair of polar coordinates for this point such that r < 0 and 0 ≤ 0 < 2TT. r = 0 =
Pair of polar coordinates for the point (-1, -π/2) satisfying the given conditions is (r, θ) = (-1, 0).
(i) To find another pair of polar coordinates for the point (3, 0) such that r > 0 and 2π ≤ θ < 4π, we can add any multiple of 2π to the angle while keeping the same value of r. Let's choose θ = 2π:
r = 3, θ = 2π
Therefore, another pair of polar coordinates for the point (3, 0) satisfying the given conditions is (r, θ) = (3, 2π).
(ii) To find another pair of polar coordinates for the point (3, 0) such that r < 0 and 0 ≤ θ < 2π, we can choose a negative value of r and add any multiple of 2π to the angle. Let's choose r = -3 and θ = 0:
r = -3, θ = 0
Therefore, another pair of polar coordinates for the point (3, 0) satisfying the given conditions is (r, θ) = (-3, 0).
(b) To find another pair of polar coordinates for the point (2, -π/7) such that r > 0 and 2π ≤ θ < 4π, we can add any multiple of 2π to the angle while keeping the same value of r. Let's choose θ = 2π:
r = 2, θ = 2π
Therefore, another pair of polar coordinates for the point (2, -π/7) satisfying the given conditions is (r, θ) = (2, 2π).
To find another pair of polar coordinates for the point (2, -π/7) such that r < 0 and -2π ≤ θ < 0, we can choose a negative value of r and add any multiple of 2π to the angle. Let's choose r = -2 and θ = -π:
r = -2, θ = -π
Therefore, another pair of polar coordinates for the point (2, -π/7) satisfying the given conditions is (r, θ) = (-2, -π).
(c) To find another pair of polar coordinates for the point (-1, -π/2) such that r > 0 and 2π ≤ θ < 4π, we can add any multiple of 2π to the angle while keeping the same value of r. Let's choose θ = 2π:
r = -1, θ = 2π
Therefore, another pair of polar coordinates for the point (-1, -π/2) satisfying the given conditions is (r, θ) = (-1, 2π).
To find another pair of polar coordinates for the point (-1, -π/2) such that r < 0 and 0 ≤ θ < 2π, we can choose a negative value of r and add any multiple of 2π to the angle. Let's choose r = -1 and θ = 0:
r = -1, θ = 0
Therefore, another pair of polar coordinates for the point (-1, -π/2) satisfying the given conditions is (r, θ) = (-1, 0).
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One year Josh had the lowest ERA (earned-run average, mean number of runs yielded per nine innings pitched) of any male pitcher at his school, with an ERA of 2.56. Also, Beth had the lowest ERA of any female pitcher at the school with an ERA of 2.74. For the males, the mean ERA was 4.747 and the standard deviation was 0.927. For the females, the mean ERA was 4.285 and the standard deviation was 0.798. Find their respective z-scores. Which player had the better year relative to their peers, Josh or Beth? (Note: In general, the lower the ERA, the better the pitcher.) Josh had an ERA with a z-score of ___
Beth had an ERA with a z-score of ___
(Round to two decimal places as needed.) Which player had a better year in comparison with their peers? A. Josh had a better year because of a higher z-score. B. Josh had a better year because of a lower z-score. C. Beth had a better year because of a lower z-score. D. Beth had a better year because of a higher z-score.
Given data:Josh had an ERA of 2.56, mean ERA for males= 4.747 and standard deviation= 0.927Beth had an ERA of 2.74, mean ERA for females= 4.285 and standard deviation= 0.798The formula for calculating the z-score is;Z-score = (x-μ)/σ
Where,x = the raw scoreμ = the population meanσ = the standard deviationLet's calculate the respective z-scores for Josh and Beth.Z-score for JoshZ-score[tex]= (x-μ)/σ = (2.56 - 4.747)/0.927= -2.36[/tex]Therefore, Josh had an ERA with a z-score of -2.36.Z-score for BethZ-score[tex]= (x-μ)/σ = (2.74 - 4.285)/0.798= -1.93[/tex]Therefore, Beth had an ERA with a z-score of -1.93.In general, the lower the z-score, the better the performance.
As Josh has a lower z-score than Beth, he had a better year relative to his peers.Therefore, the correct option is B. Josh had a better year because of a lower z-score.
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Option B is correct, Josh had a better year because of a lower z-score.
To find the z-scores of Josh and Beth, we'll use the formula:
z= x−μ/σ
where:
x is the individual's ERA.
μ is the mean ERA of their respective group (males or females).
σ is the standard deviation of the ERA for their respective group.
For Josh:
Josh's ERA (x) = 2.56
Mean ERA for males (μ) = 4.747
Standard deviation for males (σ) = 0.927
Substituting these values into the z-score formula for Josh:
Z josh = 2.56-4.747/0.927
=-2.48
For Beth:
Beth's ERA (x) = 2.74
Mean ERA for females (μ) = 4.285
Standard deviation for females (σ) = 0.798
z Beth = 2.74-4.285/0.798
=-1.93
Now, comparing the z-scores, we can determine which player had a better year relative to their peers.
A lower z-score indicates a better performance relative to the mean.
In this case, Josh has a z-score of -2.48, while Beth has a z-score of -1.93. Since the z-score for Josh is lower (further below the mean) than the z-score for Beth, we can conclude that Josh had a better year relative to his peers.
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Find the margin of error for the given values of c, d, and n. c-0.95, 0-677, n-40 Question 7 Provide an appropriate response. For a sample of 20 IQ scores the mean score is 105.8. The standard deviation, a, is 15. Determine whether a normal distribution or at-distribution should be used or whether neither of these can be used to construct a confidence interval. Assume that IQ scores are normally distributed
In this scenario, a normal distribution can be used to construct a confidence interval for the IQ scores, assuming that the IQ scores are normally distributed.
To determine whether a normal distribution or t-distribution should be used to construct a confidence interval for a sample of IQ scores, we need to consider the sample size and whether the population standard deviation is known or unknown.
In this case, we are given a sample size of 20 and the standard deviation of the population (a) is known to be 15. Since the population standard deviation is known, we can use a normal distribution to construct a confidence interval.
When the population standard deviation is known and the sample size is relatively small (typically less than 30), the sample distribution can be approximated by a normal distribution. In such cases, using a normal distribution is appropriate for constructing confidence intervals.
Therefore, in this scenario, a normal distribution can be used to construct a confidence interval for the IQ scores, assuming that the IQ scores are normally distributed.
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A clinical trial was conducted to test the effectiveness of a drug used for treating insomnia in older subjects. After treatment with the drug. 20 subjects had a mean wake time of 97.5 min and a standard deviation of 44.1 min. Assume that the 20sample values appear to be from a normally distributed population and construct a 90% confidence interval estimate of the standard deviation of the wake times for a population with the drug treatments. Does the result indicate whether the treatment is effective?
The 90% confidence interval estimate of the standard deviation of the wake times for a population with the drug treatment is (30.77, 71.14) minutes. The result does not directly indicate whether the treatment is effective.
To construct a confidence interval estimate of the standard deviation, we can use the chi-square distribution. The formula for the confidence interval estimate of the standard deviation is:
CI = [(n - 1) * s^2 / χ^2 upper, (n - 1) * s^2 / χ^2 lower]
Where n is the sample size, s is the sample standard deviation, and χ^2 upper and χ^2 lower are the upper and lower critical values from the chi-square distribution.
In this case, with a sample size of 20, a sample standard deviation of 44.1 minutes, and a 90% confidence level, we can calculate the confidence interval estimate of the standard deviation.
Using the chi-square distribution table or a statistical software, we find that the upper critical value χ^2 upper is 32.852 and the lower critical value χ^2 lower is 9.591.
Plugging in the values into the formula, we obtain the confidence interval estimate of the standard deviation as (30.77, 71.14) minutes.
The confidence interval estimate provides a range of plausible values for the standard deviation of wake times. However, it does not directly indicate whether the treatment is effective. To determine the effectiveness of the treatment, further analysis and comparison with other groups or control conditions would be necessary. The confidence interval estimate provides a measure of the precision of the estimated standard deviation, but additional evidence and evaluation would be required to assess the effectiveness of the drug treatment for insomnia in older subjects.
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President Barnes would like to know more about students at Cuyamaca College and has formed a committee to analyze a campus-wise online survey recently done via SurveyMonkey.com. The sample consists of responses from n - 173 randomly selected students and is believed to be representative of the student body at Cuyamaca. It was noted that among 173 students in the survey, 53 participate in varsity sports. a. What is the point estimate for the proportion of Cuyamaca students who are varsity athletes? Round to 3 decimal places. b. Construct a 95% confidence interval for the proportion of Cuyamaca students who are varsity athletes. Round to 3 decimal places. c. Write a one sentence interpretation of your confidence interval
a. The point estimate for the proportion of Cuyamaca students who are varsity athletes is 0.309 (rounded to 3 decimal places).
b. The 95% confidence interval for the proportion of Cuyamaca students who are varsity athletes is (0.238, 0.380) (rounded to 3 decimal places).
c. We are 95% confident that the true proportion of Cuyamaca students who are varsity athletes is between 23.8% and 38.0%.
a. Point estimate
The point estimate is the sample proportion of Cuyamaca students who are varsity athletes. This is calculated by dividing the number of students who participate in varsity sports by the total number of students in the sample. In this case, there are 53 students who participate in varsity sports and 173 students in the sample, so the point estimate is 53 / 173 = 0.309.
b. Confidence interval
The confidence interval is a range of values that is likely to contain the true population proportion. The 95% confidence interval means that we are 95% confident that the true proportion of Cuyamaca students who are varsity athletes is between 23.8% and 38.0%. This interval is calculated using the sample proportion, the sample size, and the z-score for a 95% confidence interval.
c. Interpretation of confidence interval
The confidence interval tells us that there is a 95% chance that the true proportion of Cuyamaca students who are varsity athletes is between 23.8% and 38.0%. This means that we can be fairly confident that the true proportion is not much different from the sample proportion of 30.9%.
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While Mary Corens was a student at the University of Tennessee, she borrowed $15,000 in student loans at an annual interest rate of 9%. If Mary repays $1,800 per year, then how long (to the nearest year) will it toke her to fepay the loan? Do not round intermediate caiculations. Round your answer to the nearest whole number.
To determine how long it will take Mary Corens to repay her student loan, we can divide the total loan amount of $15,000 by the annual repayment amount of $1,800. The result will give us the number of years it will take her to repay the loan.
it will take Mary approximately 8 years to repay the loan.
By dividing the total loan amount of $15,000 by the annual repayment amount of $1,800, we can calculate the number of years needed to repay the loan.
Loan amount: $15,000
Annual repayment: $1,800
Number of years = Loan amount / Annual repayment
Number of years = $15,000 / $1,800
Number of years ≈ 8.33
Since we are asked to round the answer to the nearest whole number, Mary will take approximately 8 years to repay the loan.
It's important to note that this calculation assumes a constant annual repayment amount of $1,800 throughout the entire loan repayment period. In reality, factors such as interest accrual and varying repayment schedules may affect the actual time it takes to fully repay the loan. Additionally, any changes to the annual repayment amount would also impact the duration of the loan repayment.
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Find the perimeter of this trapezium.
Formula :
Perimeter of Trapezium = sum of all sides
AB + BD + EC + BC [Refer to the attachment]
AB = 14 cm
EC = 6 cm
BD = 8 cm
Let's find the length of side BC
In right angle triangle , BDC
EC = 14 cm
AB = 6 cm
DC = EC - AB
= 14 - 6
= 8 cm
According to Pythagoras theorem,
BC² = DC² + BD²
BC² = 8² + 8²
BC² = 64 + 64
BC = √128
BC = 11.31 cmPerimeter = sum of all sides
= 14 + 6 + 8 +11.31
= 20 + 8 + 11.31
= 28 + 11.31
= 39.21 cm (Answer)Find the marked angles in Fig. 13.25. 4x X 3x
The marked angles in Fig. 13.25 are 96 degrees and 72 degrees.
In Fig. 13.25, we have two parallel lines AB and CD. We also have a transversal XY that intersects these two parallel lines. We need to find the marked angles, which are 4x and 3x.
Step 1: Identify the pairs of corresponding angles.
The corresponding angles are the ones that are on the same side of the transversal and in the same position with respect to the parallel lines.
The corresponding angles are equal. For example, angle AXY and angle CYX are corresponding angles and are equal. Similarly, angle BYX and angle DXY are corresponding angles and are equal. We can write the corresponding angles as follows: Angle AXY = angle CYXAngle BYX = angle DXYStep 2:Identify the pairs of alternate interior angles.
The alternate interior angles are the ones that are on opposite sides of the transversal and in the same position with respect to the parallel lines.
The alternate interior angles are equal. For example, angle BXY and angle CXD are alternate interior angles and are equal. Similarly, angle AYX and angle DYC are alternate interior angles and are equal. We can write the alternate interior angles as follows:
Angle BXY = angle CXDAngle AYX = angle DYCStep 3:Identify the pair of interior angles on the same side of the transversal. The interior angles on the same side of the transversal are supplementary. That is, their sum is 180 degrees.
For example, angle AXY and angle BYX are interior angles on the same side of the transversal, and their sum is 180 degrees. We can write this as follows: Angle AXY + angle BYX = 180Step 4:Use the relationships we have identified to solve for x.
We can start by using the relationship between angle BXY and angle CXD, which are alternate interior angles. We have angle BXY = angle CXD4x = 3x + 10x = 10Next, we can use the relationship between angle AXY and angle BYX, which are interior angles on the same side of the transversal.
We have:angle AXY + angle BYX = 180(3x + 10) + 4x = 1807x + 10 = 1807x = 170x = 24Finally, we can substitute x = 24 into the expressions for 4x and 3x to find the marked angles. We have:4x = 4(24) = 963x = 3(24) = 72Therefore, the marked angles in Fig. 13.25 are 96 degrees and 72 degrees.
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(1 point) Given the curve R(t) = 2 sin(2t) i + 2 cos(2t)j + 4k (1) Find R' (t) (2) Find R" (t) = = (3) Find the curvature * =
The second derivative of the given curve is obtained as follows:R"(t) = (-8sin(2t) i - 8cos(2t)j + 0k)(3) .
The curvature of the given curve is obtained as follows:curvature k(t) = (||R'(t)×R"(t)||)/ ||R'(t)||³Putting in the values of R'(t) and R"(t), we get:k(t) = ((16(cos²(2t) + sin²(2t)))^(1.5))/16 = 1
Given the curve R(t) = 2sin(2t) i + 2cos(2t) j + 4k, we need to find the following:(1) R'(t)(2) R"(t)(3) Curvature
The first derivative of the given curve is obtained by differentiating each component of the curve with respect to t.Using the formula of differentiation of sine and cosine functions, we get:
R'(t) = (d/dt)(2sin(2t) i + 2cos(2t) j + 4k) = (4cos(2t) i - 4sin(2t)j + 0k)
Therefore, R'(t) = 4cos(2t) i - 4sin(2t)j
The second derivative of the given curve is obtained by differentiating R'(t) with respect to t.
Similarly, we get:
R"(t) = (d/dt)(R'(t)) = (-8sin(2t) i - 8cos(2t)j + 0k)
Therefore, R"(t) = -8sin(2t) i - 8cos(2t)j
The curvature of a curve is defined as the rate at which its tangent rotates with respect to its arc length. It is given by the formula:
k(t) = (||R'(t)×R"(t)||)/ ||R'(t)||³
The magnitude of the cross product of R'(t) and R"(t) is given by:
||R'(t)×R"(t)|| = ||(4cos(2t) i - 4sin(2t)j)×(-8sin(2t) i - 8cos(2t)j)||= ||(32cos(2t) - 32sin(2t))k||= 32||(cos(2t) - sin(2t))k||= 32(√2)/2= 16√2
The magnitude of the first derivative is given by:||R'(t)|| = √(16cos²(2t) + 16sin²(2t))= 4
Therefore, the curvature of the given curve is:k(t) = (||R'(t)×R"(t)||)/ ||R'(t)||³= (16√2)/64= √2/4
Therefore, the main answer for the given problem is as follows:(1) R'(t) = 4cos(2t) i - 4sin(2t)j(2) R"(t) = -8sin(2t) i - 8cos(2t)j(3) The curvature of the given curve is k(t) = √2/4
The given problem is related to the first and second derivative of a curve and the calculation of its curvature. The solutions to the three parts of the problem are obtained by using the formulas of differentiation, cross product, and magnitude. The main answer is summarized in the final paragraph of the solution.
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Home Depot wants to know how long the average light bulb lasts. They sample 3,500 families in the United States. Based on past studies the population standard deviation is assumed known to be 400 hours. You calculate the sample mean to be 1,800 hours. What is the correct interpretation of the 98% confidence interval estimate? You are 98% confident that the population mean hours a light bulb lasts in the United States is within the interval 1,784 hours to 1,815 hours The population mean hours a light bulb lasts in the United States will be within the interval 1,784 hours to 1,815 hours 98% of the time You are 98% confidence that the hours a light bulb lasts in the United States is within the interval 1,784 hours to 1,815 hours You are 98% confident that the sample mean hours a light bulb lasts in the United States is within the interval 1,784 hours to 1,815 hours
We are 98% confident that the true population mean of the lifespan of light bulbs falls between 1,784.63 hours and 1,815.37 hours.
The 98% confidence interval estimate for the population mean lifespan of light bulbs in the United States, based on a sample of 3,500 families, is between 1,784 hours and 1,815 hours.
According to the question,
We can calculate a 98% confidence interval for the average lifespan of a light bulb based on the sample mean of 1,800 hours and a population standard deviation of 400 hours.
Using a standard formula and the given data,
We can calculate the margin of error to be approximately 28.62 hours. This means that we can be 98% confident that the true average lifespan of a light bulb falls within the range of 1,771.38 hours to 1,828.62 hours.
Therefore, the correct interpretation of the 98% confidence interval estimate is that we are highly confident that the true average lifespan of a light bulb for the population falls within this range,
Based on the sample data collected from 3,500 families in the United States.
The correct interpretation of a 98% confidence interval is that we are 98% confident that the true population mean falls within the range of 1,784 hours to 1,815 hours based on the sample data we collected from the 3,500 families in the United States.
It's important to note that this confidence interval estimate provides a range of values within which the true population mean is likely to fall. It does not mean that the true population mean is necessarily within this range with 98% certainty. Rather, it means that if we were to repeat this study many times and construct 98% confidence intervals using the same method, 98% of the intervals would contain the true population mean.
Therefore, we cannot say that the population mean hours a light bulb lasts in the United States will be within the interval 1,784 hours to 1,815 hours 98% of the time.
Rather, we can say that there is a 98% chance that the true population mean falls within this interval based on the sample data we collected.
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Which of these expressions is equivalent to 30b2?
A 3b + 10b
B 3b. 10b
c9b +21b
D 9b21b
Answer:
b
Step-by-step explanation:
A. you can just sum the two terms containing a "b"
3b + 10b = 13b
B. can be writen like 3 × b × 10 × b
3b. 10b = 3×b×10×b = 30×b×b = 30b²
c. same as in A.
9b +21b = 30b
D. is any of those numbers a power? if not, it's the same as in B.
9b21b = 9×b×21×b = 189b²
How many students must be randomly selected to estimate the mean monthly income of students at a university? Suppose we want 95% confidence that X is within $135 of μ, and the o is known to be $549. O A. 110 OB. 63 OC. 549 OD. 0 O E. 135 OF. 45 O G. none of the other answers O H. 7 G
How many students must be randomly selected to estimate the mean monthly income of students at a university, given that we want 95% confidence that X is within $135 of μ.
and the o is known to be $549?To determine the number of students that should be chosen, we'll use the margin of error formula, which is: E = z (o / √n) where E represents the margin of error, z represents the critical value, o represents the population standard deviation, and n represents the sample size.
Since we want to be 95% confident that the sample mean is within $135 of the true population mean, we can write this as: Z = 1.96 (from the standard normal table)
E = $135o
= $549
Plugging these values into the formula: E = z (o / √n)$135
= 1.96 ($549 / √n)$135 / 1.96
= $549 / √n68.88 = $549 / √nn
= ($549 / $68.88)^2n
≈ 63 Therefore, we need to randomly select at least 63 students to estimate the mean monthly income of students at a university with 95% confidence that the sample mean is within $135 of the true population mean.
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A study of consumer smoking habits includes 200 people in the 18-22 age bracket ( 40 of whom smoke), 130 people in the 23-30 age bracket (31 of whom smoke), and 100 people in the 31−40 age bracket ( 30 of Whom smoke). If one person is randomly selected from this sample, find the probability of getting someone who is age 18−22 or smokes.
a. 0.593 b. 0.607 c. 0.695 d. 0.468
The probability of randomly selecting someone from the given sample who is either in the age bracket of 18-22 or smokes is 0.607 (option b).
To calculate this probability, we need to consider the total number of individuals in the sample who are either in the age bracket of 18-22 or smoke, and divide it by the total number of individuals in the sample.
Step 1: Calculate the number of individuals in the sample who are in the age bracket of 18-22 or smoke.
- Number of individuals in the age bracket of 18-22 = 200
- Number of individuals in the age bracket of 18-22 who smoke = 40
- Number of individuals in the age bracket of 23-30 = 130
- Number of individuals in the age bracket of 23-30 who smoke = 31
- Number of individuals in the age bracket of 31-40 = 100
- Number of individuals in the age bracket of 31-40 who smoke = 30
Total number of individuals in the sample who are either in the age bracket of 18-22 or smoke = (Number of individuals in the age bracket of 18-22) + (Number of individuals in the age bracket of 23-30 who smoke) + (Number of individuals in the age bracket of 31-40 who smoke)
= 200 + 31 + 30
= 261
Step 2: Calculate the total number of individuals in the sample.
- Total number of individuals in the age bracket of 18-22 = 200
- Total number of individuals in the age bracket of 23-30 = 130
- Total number of individuals in the age bracket of 31-40 = 100
Total number of individuals in the sample = (Total number of individuals in the age bracket of 18-22) + (Total number of individuals in the age bracket of 23-30) + (Total number of individuals in the age bracket of 31-40)
= 200 + 130 + 100
= 430
Step 3: Calculate the probability.
- Probability = (Number of individuals in the sample who are either in the age bracket of 18-22 or smoke) / (Total number of individuals in the sample)
= 261 / 430
= 0.607
Therefore, the probability of randomly selecting someone from the given sample who is either in the age bracket of 18-22 or smokes is 0.607 (option b).
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Question Let h(x) = f(x) . If f(x) = -2x²-7. g(-2) = -7, and g'(-2) = 5. Find h'(-2). g(x) Provide your answer below: h'(-2)=
To find h'(-2), we first need to find h'(x), the derivative of h(x).We haveh(x) = f(x)g(x). Using the product rule for derivatives, we get:
h'(x) = f'(x)g(x) + f(x)g'(x)
Therefore, h'(-2) = f'(-2)g(-2) + f(-2)g'(-2)
Now, we are given that f(x) = -2x² - 7 and g(-2) = -7 and g'(-2) = 5.
We first find f'(-2), the derivative of f(x) at x = -2.
Using the power rule for derivatives, we get:
f'(x) = -4xTherefore, f'(-2) = -4(-2) = 8
Now we substitute the values in the formula we derived above:
h'(-2) = f'(-2)g(-2) + f(-2)g'(-2)= 8(-7) + (-2(-2)² - 7)(5)= -56 + (-2(4) - 7)(5)= -56 + (-8 - 7)(5)= -56 - 75= -131
Therefore, h'(-2) = -131.
Therefore, h'(-2) = -131.
The derivative of h(x) at x = -2 is h'(-2) = -131.
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A baseball player has a lifetime batting average of 0.163. If, in a season, this player has 300 "at bats", what is the probability he gets 40 or more hits? Probability of 40 or more hits =
The probability he gets 40 or more hits is 0.9154.
Given the batting average of a baseball player in his lifetime is 0.163, and in a season he has 300 at-bats, to determine the probability he gets 40 or more hits, let's proceed as follows;
The mean is calculated as follows:μ = npμ = 300 x 0.163μ = 48.9.
The variance is calculated as follows:σ2 = npqσ2 = 300 x 0.163 x (1 - 0.163)σ2 = 300 x 0.137887σ2 = 41.3661.
Standard deviation (SD) is calculated as follows:σ = √(300 x 0.163 x (1 - 0.163))σ = √41.3661σ = 6.4309z-score is calculated as follows:z = (X - μ) / σWhere X = 40z = (40 - 48.9) / 6.4309z = -1.377.
Probability of getting 40 or more hits is calculated as follows:P(X ≥ 40) = P(Z ≥ -1.377)P(Z ≥ -1.377) = 1 - P(Z < -1.377).
Using a z-score table;the area to the left of z = -1.37 is 0.0846P(Z ≥ -1.377) = 1 - 0.0846P(Z ≥ -1.377) = 0.9154.
Therefore, the probability he gets 40 or more hits is 0.9154. The main answer is: The probability he gets 40 or more hits is 0.9154.
A baseball player has a lifetime batting average of 0.163. If, in a season, this player has 300 "at bats", the probability he gets 40 or more hits can be calculated as follows:To determine the probability he gets 40 or more hits, we need to find the mean, variance, and standard deviation of his hits in 300 at-bats.
First, the mean is calculated as μ = np.μ = 300 x 0.163μ = 48.9. The variance is calculated as σ2 = npq.σ2 = 300 x 0.163 x (1 - 0.163).σ2 = 300 x 0.137887σ2 = 41.3661.
Standard deviation (SD) is calculated as σ = √(300 x 0.163 x (1 - 0.163)).σ = √41.3661σ = 6.4309.Now, we need to find the z-score of getting 40 or more hits.
The z-score is calculated as z = (X - μ) / σ, where X = 40. z = (40 - 48.9) / 6.4309. z = -1.377.
The probability of getting 40 or more hits is calculated as P(X ≥ 40) = P(Z ≥ -1.377) = 1 - P(Z < -1.377). Using a z-score table, the area to the left of z = -1.37 is 0.0846. P(Z ≥ -1.377) = 1 - 0.0846 = 0.9154.
Therefore, the probability he gets 40 or more hits is 0.9154.
In conclusion, the probability of the baseball player getting 40 or more hits in a season is 0.9154.
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Find the values of the constant c which makes the function continuous on the [cx¹ +7cxª³+2, x < -1 interval (-[infinity]0,00): f(x) = Ac-2² -cr. T>1
For any value of c, the equation holds true, meaning the function is continuous at x = -1 for all values of c. Any value of c will make the function continuous on the interval (−∞, 0) and (0, ∞).
To find the values of the constant c that make the function continuous on the interval (−∞, 0) and (0, ∞), we need to ensure that the left-hand limit and the right-hand limit of the function are equal at the point of discontinuity, which is x = -1.
First, let's find the left-hand limit of the function as x approaches -1. We substitute x = -1 into the function:
lim(x → -1-) f(x) = lim(x → -1-) (Ac^2 - cr)
Next, let's find the right-hand limit of the function as x approaches -1:
lim(x → -1+) f(x) = lim(x → -1+) (Ac^2 - cr)
To make the function continuous at x = -1, the left-hand limit and the right-hand limit must be equal:
lim(x → -1-) f(x) = lim(x → -1+) f(x)
Now, let's evaluate the left-hand and right-hand limits:
lim(x → -1-) (Ac^2 - cr) = lim(x → -1+) (Ac^2 - cr)
Simplifying the expressions:
(Ac^2 - c(-1)) = (Ac^2 - c(-1))
Ac^2 + c = Ac^2 + c
The c terms cancel out, leaving:
Ac^2 = Ac^2
We can see that for any value of c, the equation holds true, meaning the function is continuous at x = -1 for all values of c.
Therefore, any value of c will make the function continuous on the interval (−∞, 0) and (0, ∞).
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what’s the answer ???
The reasons for the steps are ;
step1 ; collect like terms
step2 : dividing both sides by 6
What is linear equation?A linear equation is an algebraic equation of the form y=mx+b. involving only a constant and a first-order (linear) term, where m is the slope and b is the y-intercept.
For example , in an equation 6x +5x = 3x + 24 , to find x in this equation we need to follow some steps;
First we collect like terms
6x +5x - 3x = 24
8x = 24
then we divide both sides by the coefficient of x
x = 24/8
x = 3
Similarly , solving 18 - 2x = 4x
collect like terms
18 = 4x +2x
18 = 6x
divide both sides by coefficient of x
x = 18/6 = 3
x = 3
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Suppose that f(t) is periodic with period [-, π) and has the following complex Fourier coefficients: ... co= -4, C₁ = 1-3i, C₂ = −4-3i, c3 = 4i, (A) Compute the following complex Fourier coefficients. C-3-4₁₁ C_2 = -4+3i, C-1 = 1+3i (B) Compute the real Fourier coefficients. (Remember that ei kt = cos(kt) + i sin(kt).) ao = -8, a₁ = 2, a2 = -8, a3 = 0 b₁ = 6, b₂ = 6 b3 = -8 (C) Compute the complex Fourier coefficients of the following. (i) The derivative f'(t). Co = 0, C₁ = i+3 C2 -8i+6 C3 = -12 (ii) The shifted function f(t +) Co= -4 C1 = (1-3i)*(1/2+i*sqrt3/2 C2 = (-4-3i)*(1/2+i*sqrt3/2, -4i (iii) The function f(3t). Co = 4, C₁ = 0 3 -4i 1 C3 = C2 = 0 || 3
(A) Computing the complex Fourier coefficients: C₋₃ = -4, C₂ = -4 + 3i, C₋₁ = 1 + 3i. (B) Computing the real Fourier coefficients: a₀ = -8, a₁ = 2, a₂ = -8, a₃ = 0, b₁ = 6, b₂ = 6, b₃ = -8
(C) Computing the complex Fourier coefficients of the following:
(i) The derivative f'(t):
C₀ = 0
C₁ = i + 3
C₂ = -8i + 6
C₃ = -12
(ii) The shifted function f(t + π):
C₀ = -4
C₁ = (1 - 3i) * (1/2 + i√3/2)
C₂ = (-4 - 3i) * (1/2 + i√3/2)
C₃ = -4i
(iii) The function f(3t):
C₀ = 4
C₁ = 0
C₂ = 3 - 4i
C₃ = 1
C₄ = 0
(A) The complex Fourier coefficients for the given function are as follows:
C₋₃ = -4, C₂ = -4 + 3i, C₋₁ = 1 + 3i. These coefficients represent the complex amplitudes of the corresponding frequency components in the Fourier series representation of the periodic function.
(B) The real Fourier coefficients can be computed from the complex coefficients:
a₀ = -8, a₁ = 2, a₂ = -8, a₃ = 0, b₁ = 6, b₂ = 6, b₃ = -8. The real coefficients are derived by separating the complex coefficients into their real and imaginary parts.
(C) Computing the complex Fourier coefficients of the derivative f'(t) yields: C₀ = 0, C₁ = i + 3, C₂ = -8i + 6, C₃ = -12. The derivative introduces a phase shift and changes the amplitudes of the frequency components.
For the shifted function f(t + π), the complex Fourier coefficients are: C₀ = -4, C₁ = (1 - 3i) * (1/2 + i√3/2), C₂ = (-4 - 3i) * (1/2 + i√3/2), C₃ = -4i. The shift affects the phase angles of the coefficients.
For the function f(3t), the complex Fourier coefficients are: C₀ = 4, C₁ = 0, C₂ = 3 - 4i, C₃ = 1, C₄ = 0. The function f(3t) introduces a change in frequency, resulting in different coefficient values.
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Calculate the derivative of the function y= (x²+3)(x-1)² x4(x³+5)³ without using Quotient Rule. No credits will be given if you use Quotient Rule. Do not simplify your answer.
The derivative of the function y = (x²+3)(x-1)² x⁴(x³+5)³ without using the Quotient Rule is calculated by applying the Product Rule and the Chain Rule.
The derivative involves multiple steps, combining the derivatives of each term while considering the chain rule for the nested functions.
To find the derivative of the given function, we can apply the Product Rule and the Chain Rule. Let's break down the function into its individual terms: (x²+3), (x-1)², x⁴, and (x³+5)³.
Using the Product Rule, we can calculate the derivative of the product of two functions. Let's denote the derivative of a function f(x) as f'(x).
The derivative of (x²+3) with respect to x is 2x, and the derivative of (x-1)² is 2(x-1). Applying the Product Rule, we get:
[(x²+3)(2(x-1)) + (x-1)²(2x)] x⁴(x³+5)³
Next, we differentiate x⁴ using the power rule, which states that the derivative of xⁿ is nxⁿ⁻¹. Hence, the derivative of x⁴ is 4x³.
For the term (x³+5)³, we need to use the Chain Rule. The derivative of the outer function (u³) with respect to u is 3u². The derivative of the inner function (x³+5) with respect to x is 3x². Therefore, applying the Chain Rule, the derivative of (x³+5)³ is 3(x³+5)² * 3x².
Combining all the derivatives, we get the final result:
[2x(x²+3)(x-1)² + 2(x-1)²(2x)] x⁴(x³+5)³ + 4x³(x²+3)(x-1)² x³(x³+5)² * 3x².
This expression represents the derivative of the function y = (x²+3)(x-1)² x⁴(x³+5)³ without using the Quotient Rule.
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You wish to test the following claim (H_{a}) at a significance level of alpha = 0.01 .
H : mu = 89.5 H_{a} / mu > 89.5
You believe the population is normally distributed, but you do not know the standard deviation. You obtain a sample of size n = 300 with mean M = 91.6 and a standard deviation of SD = 10.6
What is the test statistic for this sample? (Report answer accurate to three decimal places.)
test statistic-
What is the p-value for this sample? (Report answer accurate to four decimal places.)
p-value-
The p-value is...
Oless than (or equal to) a
Ogreater than a
This test statistic leads to a decision to...
O reject the null
O accept the null
O fail to reject the null
The test statistic for the sample is approximately 3.295. The p-value for this sample is approximately 0.0006. The p-value is less than the significance level of 0.01. Therefore, based on the test statistic and p-value, the decision is to reject the null hypothesis.
To test the claim that the population mean (μ) is greater than 89.5 at a significance level of α = 0.01, we can perform a one-sample t-test. Since the population standard deviation (σ) is unknown, we will use the sample standard deviation (SD = 10.6) as an estimate.
The test statistic for this sample is calculated using the formula:
t = (M - μ) / (SD / √n)
Plugging in the values from the problem, we have:
t = (91.6 - 89.5) / (10.6 / √300) ≈ 3.295
The p-value for this sample can be found by comparing the test statistic to the t-distribution with n - 1 degrees of freedom. Since the alternative hypothesis is μ > 89.5, we are interested in the right-tail area.
Using statistical software or a t-table, we find that the p-value associated with a t-statistic of 3.295 and 299 degrees of freedom is approximately 0.0006.
Comparing the p-value to the significance level (α = 0.01), we can see that the p-value (0.0006) is less than α. Therefore, the p-value is less than or equal to α.
This test statistic leads to a decision to reject the null hypothesis. In other words, there is sufficient evidence to support the claim that the population mean is greater than 89.5 at a significance level of 0.01.
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for the bypotheris test to p-valne 0.101. using a level of significanse of a 0 os, determine if we reject or fal to roject the nall hypotheris. Fail to reject the aull. There as safficient evidence to eenchade oew procedure decreases prodoction time. Rejoct the null. There is imsufficient evideoce to coeclude the nets pecocobare dereases production time. Reject the mall. There is sutficient evidence to cooclude the now procthure decterses productica time. Fail to reject the null. There is innufficieat evidence to conchisde the new procodure decreaser production tima.
Given that the p-value of the hypothesis test is 0.101 and the level of significance is α = 0.05. We are to determine if we reject or fail to reject the null hypothesis.
Therefore, the decision rule is: Reject the null hypothesis if the p-value is less than or equal to the level of significance.Fail to reject the null hypothesis if the p-value is greater than the level of significance. Since the p-value 0.101 > 0.05 (level of significance), we fail to reject the null hypothesis.
Thus, the correct statement is: Fail to reject the null. There is insufficient evidence to conclude the new procedure decreases production time. Therefore, the decision rule is: Reject the null hypothesis if the p-value is less than or equal to the level of significance. Fail to reject the null hypothesis if the p-value is greater than the level of significance.
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Vignette C Suppose Toyota wants to study how many TV viewers recall the TV commercials for its newest Toyota Prius model. Someone on the marketing team claims that "a sample of 800 viewers is always better than a sample of 400 viewers. Period." Do you agree or disagree with this statement? Explain your reasoning.
I disagree with the statement that "a sample of 800 viewers is always better than a sample of 400 viewers. Period." The sample size is not the only factor that determines the quality of a study.
The sample size is important because it determines the precision of the results. A larger sample size will lead to more precise results, meaning that the confidence interval will be narrower. However, a larger sample size is not always necessary or even desirable. If the sample is not representative of the population, then even a large sample size may not be accurate. Additionally, if the data is collected in a biased way, then even a large sample size may not be reliable.
In the case of Toyota's study, the sample size of 800 viewers may be overkill. If the sample is representative of the population of TV viewers, then a sample size of 400 viewers may be sufficient to produce accurate results. However, if Toyota is interested in a specific subgroup of TV viewers, such as Prius owners, then a larger sample size may be necessary to ensure that the results are accurate.
The decision of how large a sample size to use should be made based on a number of factors, including the precision of the results desired, the representativeness of the sample, and the way the data is collected.
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The curve y = ax³ + bx² + cx+d has a critical point at (-1,3) and has a point of inflection at (0.1). Find the equation of the curve that will make the conditions true. Oy=-x³+4x² + 3x-2 Oy=x²+x-7 O y=-x³ + 3x² + 2x O y = x³ - 3x + 1
the correct equation of the curve is:
y = ax³ - 3ax + 3 - 3a
To find the equation of the curve that satisfies the given conditions, we can use the information about critical points and points of inflection.
Given that the curve has a critical point at (-1,3), we know that the derivative of the curve at that point is zero. Taking the derivative of the curve equation, we have:
y' = 3ax² + 2bx + c
Substituting x = -1 and y = 3 into this equation, we get:
0 = 3a + 2b + c (Equation 1)
Next, given that the curve has a point of inflection at (0,1), we know that the second derivative of the curve at that point is zero. Taking the second derivative of the curve equation, we have:
y'' = 6ax + 2b
Substituting x = 0 and y = 1 into this equation, we get:
0 = 2b (Equation 2)
Since b = 0, we can substitute this value into Equation 1 to solve for a and c:
0 = 3a + c (Equation 3)
From Equation 2, we have b = 0, and from Equation 3, we have c = -3a.
Substituting these values into the curve equation, we have:
y = ax³ + 0x² - 3ax + d
Simplifying, we get:
y = ax³ - 3ax + d
To find the value of d, we can substitute the coordinates of one of the given points (either (-1,3) or (0,1)) into the equation.
Let's substitute (-1,3):
3 = a(-1)³ - 3a(-1) + d
3 = -a - (-3a) + d
3 = -a + 3a + d
3 = 3a + d
Simplifying, we get:
d = 3 - 3a
So the equation of the curve that satisfies the given conditions is:
y = ax³ - 3ax + (3 - 3a)
Simplifying further, we have:
y = ax³ - 3ax + 3 - 3a
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4. Suppose each of the following statements is true Lisa is a sophomore Lisa got an A in the combinatorics test or Ben got an A in the combinatorics test. • If Ben got an A on the combinatorics test
The given statements are consistent and can all be true simultaneously. we can conclude that the statements are consistent and there is no contradiction.
The given statements are:
1. Lisa is a sophomore.
2. Lisa got an A in the combinatorics test or Ben got an A in the combinatorics test.
3. If Ben got an A on the combinatorics test, then Lisa is not a sophomore.
We need to determine if the given statements are consistent or if there is a contradiction.
Let's analyze the statements:
Statement 1: Lisa is a sophomore.
This statement provides information about Lisa's academic standing.
Statement 2: Lisa got an A in the combinatorics test or Ben got an A in the combinatorics test.
This statement states that either Lisa or Ben got an A in the combinatorics test.
Statement 3: If Ben got an A on the combinatorics test, then Lisa is not a sophomore.
This statement establishes a relationship between Ben's performance in the test and Lisa's academic standing.
Based on the given information, we can conclude that the statements are consistent and there is no contradiction. Here's why:
- If Lisa is a sophomore and the second statement is true, it means that either Lisa or Ben got an A in the combinatorics test. Since Lisa is a sophomore, Ben must have received the A.
- Statement 3 states that if Ben got an A, then Lisa is not a sophomore. Since Ben got an A, Lisa cannot be a sophomore.
Therefore, the given statements are consistent and can all be true simultaneously.
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A study examines people's television watching time. The researchers found that adults spend on average 10.4 hours watching TV with a standard deviation of 4.8 hours. What percentage of adults spent between 10 and 20 hours watching TV each week?
The z-scores for 10 and 20 hours can be calculated as follows:[tex]z1=(10-10.4)/4.8=-0.0833z2=(20-10.4)/4.8=1.9583[/tex]From the normal distribution table, we can find the probabilities corresponding to the calculated z-scores.
The probability for z1 is[tex]P(z < -0.0833) = 0.4664[/tex]. Similarly,
the probability for [tex]z2 is P(z < 1.9583) = 0.9744[/tex].The percentage of adults spent between 10 and 20 hours watching TV each week can be calculated as follows[tex]:P(-0.0833 < z < 1.9583) = P(z < 1.9583) - P(z < -0.0833) = 0.9744 - 0.4664 = 0.5080 or 50.80%[/tex] (rounded off to two decimal places).Therefore, approximately 50.80% of adults spent between 10 and 20 hours watching TV each week.
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A researcher believen that 48% of people who grew up as the only child have an IQ score over 100 . However, unknown to the researcher, this figure is actually 5046, which is the same as in the general population. To attempk to find evidence for the claim, the researcher is going to take a random sample of 400 people who grew up as the only childi tet p^ be the proportion of people in the sample with an 19 seure above 100 . Answer the folsowing. (th necessary, constit a laz of tormilas.). (a) Find the mean of p (b) Find the standard devaticn of p. (c) Compite an appreximation for P(p^≥0.48), which is the probabilify that thete will be 48% or more people with tQ scores over 100 in the sample. Round your aniswer to four decimal places.
(a) The mean of p is 0.48, which represents the expected proportion of people in the sample with an IQ score above 100.
(b) The standard deviation of p is approximately 0.0244, calculated using the formula sqrt((p * (1 - p)) / n), where p is 0.48 and n is 400.
(a) The mean of p is calculated directly as p, which in this case is 0.48. This means that on average, 48% of the sample population is expected to have an IQ score above 100.
(b) The standard deviation of p can be calculated using the formula sqrt((p * (1 - p)) / n), where p is 0.48 (the proportion of interest) and n is the sample size, which is 400 in this case. Plugging in these values, we get sqrt((0.48 * (1 - 0.48)) / 400) ≈ 0.0244. The standard deviation measures the spread or variability of the proportion p in the sample.
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Actual sales for a product was 83 in June, 81 in July, and 74 in August.
If sales forecast for July is 84, what is the exponential smoothing forecast (with α=0.2α=0.2) for September?
Do not round your results.
Forecast for September =
The exponential smoothing forecast (with α=0.2) for September is 77.088.
To apply exponential smoothing with α=0.2 and forecast the sales for September, we need to first calculate the smoothed values for June, July, and August.
The smoothed value for June is equal to the actual sales in June. That is, S(June) = 83.
For July, we use the formula:
S(July) = α × Actual Sales (July) + (1 - α) × S(June)
= 0.2 × 81 + 0.8 × 83
= 81.2
So the smoothed value for July is 81.2.
Similarly, for August, we use the formula:
S(August) = α × Actual Sales (August) + (1 - α) × S(July)
= 0.2 × 74 + 0.8 × 81.2
= 78.36
So the smoothed value for August is 78.36.
Now, we can use the formula for exponential smoothing to forecast the sales in September:
Forecast for September = α × Actual Sales (August) + (1 - α) × S(August)
= 0.2 × 74 + 0.8 × 78.36
= 77.088
Therefore, the exponential smoothing forecast (with α=0.2) for September is 77.088.
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A particular fruit's weights are normally distributed, with a mean of 382 grams and a standard deviation of 40 grams. If you pick 16 fruits at random, what is the probability that their mean weight will be between 371 grams and 377 grams. Enter your answers as numbers accurate to 4 decimal places.
The probability that the mean weight of the 16 fruits will be between 371 grams and 377 grams is approximately 0.1728 (rounded to 4 decimal places).
We have,
The mean weight of the fruit population is 382 grams, and the standard deviation is 40 grams.
Since we are sampling 16 fruits at random, we are interested in the distribution of the sample means.
The distribution of the sample means will also be normally distributed, with the same mean as the population mean (382 grams) and a standard deviation equal to the population standard deviation divided by the square root of the sample size. In this case, the sample size is 16.
The standard deviation of the sample mean
= 40 grams / √(16)
= 40 grams / 4 = 10 grams.
Now, we need to find the probability that the mean weight of the 16 fruits falls between 371 grams and 377 grams.
To do this, we will convert these values to z-scores using the formula:
z = (x - mean) / standard deviation
For 371 grams:
z1 = (371 - 382) / 10
For 377 grams:
z2 = (377 - 382) / 10
Now, we can use a standard normal distribution table or a calculator to find the corresponding probabilities associated with these z-scores.
The probability that the mean weight of the 16 fruits is between 371 grams and 377 grams can be calculated as the difference between the cumulative probabilities corresponding to z1 and z2.
P(371 < x < 377) = P(z1 < z < z2)
Let's calculate the z-scores and find the probability using a standard normal distribution table or calculator.
z1 = (371 - 382) / 10 ≈ -1.1
z2 = (377 - 382) / 10 ≈ -0.5
Using the standard normal distribution table or calculator, we find the probabilities associated with z1 and z2:
P(z < -1.1) ≈ 0.1357
P(z < -0.5) ≈ 0.3085
To find the probability between z1 and z2, we subtract the cumulative probability corresponding to z1 from the cumulative probability corresponding to z2:
P(z1 < z < z2) = P(z < z2) - P(z < z1)
P(-1.1 < z < -0.5) ≈ 0.3085 - 0.1357 ≈ 0.1728
Therefore,
The probability that the mean weight of the 16 fruits will be between 371 grams and 377 grams is approximately 0.1728 (rounded to 4 decimal places).
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A simple random sample of front-seat occupants involved in car crashes is obtained. Among
2799
occupants not wearing seat belts,
32
were killed. Among
7747
occupants wearing seat belts,
19
were killed. Use a
0.01
significance level to test the claim that seat belts are effective in reducing fatalities. Complete parts (a) through (c) below.
Question content area bottom
Part 1
a. Test the claim using a hypothesis test.
Consider the first sample to be the sample of occupants not wearing seat belts and the second sample to be the sample of occupants wearing seat belts. What are the null and alternative hypotheses for the hypothesis test?
A.
H0:
p1=p2
H1:
p1≠p2
B.
H0:
p1=p2
H1:
p1
C.
H0:
p1≤p2
H1:
p1≠p2
D.
H0:
p1≥p2
H1:
p1≠p2
E.
H0:
p1≠p2
H1:
p1=p2
F.
H0:
p1=p2
H1:
p1>p2
Your answer is correct.
Part 2
Identify the test statistic.
z=enter your response here
(Round to two decimal places as needed.)
The null and alternative hypotheses for the hypothesis test are:
A. H0: p1 = p2
H1: p1 ≠ p2
In this hypothesis test, where we are comparing two proportions, the test statistic used is the z-statistic. The formula for the z-statistic is:
z = (p1 - p2) / sqrt((p(1 - p) / n1) + (p(1 - p) / n2))
where p1 and p2 are the sample proportions, p1 and p2 are the estimated population proportions, n1 and n2 are the sample sizes of the two groups.
In this case, we have p1 = 32/2799, p2 = 19/7747, n1 = 2799, and n2 = 7747. Plugging these values into the formula, we can calculate the z-statistic.
z = ((32/2799) - (19/7747)) / sqrt(((32/2799)(1 - 32/2799) / 2799) + ((19/7747)(1 - 19/7747) / 7747))
Calculating the numerator and denominator separately:
Numerator: (32/2799) - (19/7747) ≈ 0.001971
Denominator: sqrt(((32/2799)(1 - 32/2799) / 2799) + ((19/7747)(1 - 19/7747) / 7747)) ≈ 0.008429
Dividing the numerator by the denominator:
z ≈ 0.001971 / 0.008429 ≈ 0.234
Therefore, the test statistic (z) is approximately 0.234 (rounded to two decimal places).
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