find the linear approximation l(x) to y = f(x) near x = a for the function. f(x) = 1 x , a = 8

a) The fraction of people who survived the crash is 38.38%. The code used to find this is:```{r}load("titanic_as.RData")mean(titanic_as$Survived)```

b) The summary statistics for the variable Age are:```{r}summary(titanic_as$Age)```The mean age is 30.19 years, the median age is 27 years and the standard deviation is 14.59 years.

c) The summary statistics for the variable Age only for those passengers who survived the crash are:```{r}summary(titanic_as$Age[titanic_as$Survived == 1])```

The mean age of those who survived is 28.34 years, the median age is 28 years and the standard deviation is 15.01 years.

d) Histogram for the variable Sex:```{r}hist(titanic_as$Sex, main = "Histogram of Sex", xlab = "Sex", ylab = "Frequency", col = "purple", border = "white")``` Histogram for the variable Sex, but only for those who survived: ```{r}hist(titanic_as$Sex[titanic_as$Survived == 1], main = "Histogram of Sex for those who survived", xlab = "Sex", ylab = "Frequency", col = "purple", border = "white")```

e) In the entire ship, the proportion of males is greater than the proportion of females. However, among those who survived, the proportion of females is higher than the proportion of males. This can be seen from the two histograms.

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The required probability that the mean height of the sample is within 2 inches of the population mean is approximately 0.649.

Suppose the mean height in inches of all 9th-grade students at one high school is estimated. The population standard deviation is 6 inches. The heights of eight randomly selected students are taken. Let X be the mean height of the eight randomly selected students.

Then, X follows a normal distribution with mean μX and standard deviation σX, given by:

μX = μ = Population mean = Mean height of 9th-grade students = UnknownσX = σ/√n = 6/√8 = 2.12 inches.

Here, n = 8 is the sample size.

We need to find the probability that the mean height of the sample is within 2 inches of the population mean i.e.[tex]P(μ - 2 ≤ X ≤ μ + 2) = P((μ - μX)/σX ≤ (2 - μ + μX)/σX) - P((μ - μX)/σX ≤ (-2 - μ + μX)/σX)P(-0.94 ≤ Z ≤ 0.94) - P(Z ≤ -2.94) ≈ 0.651 - 0.002 = 0.649[/tex]

Note: Here, we have used the standard normal distribution table to calculate the probability of Z-score.

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Given,The table shows the number of cars of two different brands sold at a dealership during a certain month.

Brand

Cars Sold Coupes Sedans Brand 1 610 410Brand 2 300 90The number of coupes of brand 1 is 4 and for brand 2 is 9.Therefore, total number of coupes = 4 + 9 = 13.

The probability that the vehicle was Brand 2, given that the vehicle selected was a coupe is given by:$$\begin{aligned}P(Brand 2| Coupe) &= \frac{P(Coupe | Brand 2) * P(Brand 2)}{P(Coupe)} \\&= \frac{\frac{9}{300} * \frac{300}{610 + 300}}{\frac{13}{910}} \\&= \frac{\frac{27}{300 * 13}}{\frac{13}{910}} \\&= \frac{63}{1000} \\&= \boxed{0.063} \end{aligned}$$Therefore, the probability that the vehicle was Brand 2, given that the vehicle selected was a coupe is 0.063 (rounded to four decimal places as needed).

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The probability that at least one bike out of 7 randomly selected cases of stolen bicycles is recovered is approximately 0.918.

To find this probability, we can use the complement rule. First, we need to determine the probability that none of the 7 bikes is recovered. Given that the probability of one bike being recovered is 0.3, the probability of one bike not being recovered is 0.7. Using this, we can calculate the probability of none of the 7 bikes being recovered as follows:

P(none recovered) = P(not recovered) x P(not recovered) x ... (7 times)

P(none recovered) = 0.7 x 0.7 x ... (7 times)

P(none recovered) = 0.7^7

P(none recovered) = 0.082

Next, we can use the complement rule to find the probability that at least one bike out of the 7 is recovered:

P(at least one recovered) = 1 - P(none recovered)

P(at least one recovered) = 1 - 0.082

P(at least one recovered) = 0.918

Therefore, the probability that at least one bike out of 7 randomly selected cases of stolen bicycles is recovered is approximately 0.918, which is closest to option B (0.918).

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The necessary sample size will be greater than the one calculated in part a.

a. When the power is increased from 0.92 to 0.98, the required sample size increases. Since the sample size must be larger to achieve the same level of accuracy for a higher power value, this is the case.

Therefore, the necessary sample size will be greater than the one calculated in part a.

A high power value necessitates a larger sample size in order to achieve the same level of accuracy as a low power value.

Hence, as the power value increases, the sample size required also increases.

b. The required sample size will be greater than the one calculated in part a when a=0.01 and power=0.92.

The sample size required for hypothesis testing is inversely proportional to the square of the critical value.

When the significance level is reduced from 0.02 to 0.01, the critical value increases by a factor of approximately 1.3. As a result, the sample size increases since the required sample size is inversely proportional to the square of the critical value.

Therefore, the necessary sample size will be greater than the one calculated in part a.

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A 90% confidence interval for the proportion of participants who experience stomach upset with the experimental medication is estimated to be approximately 0.148 to 0.352.

To construct a confidence interval for the proportion, we use the formula p ± z * sqrt((p * (1 - p)) / n), where p is the observed proportion, z is the z-score corresponding to the desired confidence level, and n is the sample size.

In this case, the observed proportion is 12/50 = 0.24, the sample size is 50, and the desired confidence level is 90%.

Using the standard normal distribution, the z-score corresponding to a 90% confidence level is approximately 1.645.

Plugging these values into the formula, we calculate the margin of error as 1.645 * sqrt((0.24 * (1 - 0.24)) / 50) ≈ 0.101.

To construct the confidence interval, we subtract and add the margin of error to the observed proportion: 0.24 ± 0.101.

Therefore, the 90% confidence interval for the proportion of participants who experience stomach upset with the experimental medication is approximately 0.148 to 0.352. This means we can be 90% confident that the true proportion falls within this interval.

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To form a polynomial with the given zeros (-3, 0, 6) and degree 3, we can use the zero-product property and create a polynomial by multiplying the factors corresponding to each zero.

The factors corresponding to the zeros are:

(x - (-3)) = (x + 3) (for the zero -3)

(x - 0) = x (for the zero 0)

(x - 6) (for the zero 6)

Multiplying these factors, we get:

(x + 3) * x * (x - 6)

To ensure a leading coefficient of 3, we can multiply the entire polynomial by 3:

3 * (x + 3) * x * (x - 6)

Expanding the polynomial, we get:

3x(x + 3)(x - 6)

Therefore, a polynomial with the given zeros (-3, 0, 6) and degree 3, with integer coefficients and a leading coefficient of 3, is:

3x(x + 3)(x - 6).

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The Holding Period Return (HPR) for Alphabet Inc. (GOOG) from January 1st, 2021 to December 31st, 2021 is X%. The capital gain yield is Y% and the dividend gain yield is Z%.

The HPR calculation involves determining the overall return on an investment over a specific period.

To calculate the HPR for Alphabet Inc. (GOOG) during the given period, we need the closing prices at the beginning and end of the period.

Using the closing price of GOOG on January 1st, 2021, and December 31st, 2021, we can calculate the capital gain yield and dividend gain yield. The formula for HPR is:

HPR = (Ending Value - Beginning Value + Dividends) / Beginning Value

To calculate the capital gain yield, we use the formula:

Capital Gain Yield = (Ending Value - Beginning Value) / Beginning Value

And for the dividend gain yield, we use the formula:

Dividend Gain Yield = Dividends / Beginning Value

By plugging in the appropriate values from the stock prices and dividends, we can calculate the HPR, capital gain yield, and dividend gain yield.

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To find the maximum profit, we can look for the vertex of the parabolic function representing the profit.

The given profit function is:

[tex]Z = -3x^2 + 12x + 25[/tex]

We can see that the coefficient of the [tex]x^2[/tex] term is negative, which means the parabola opens downwards. This indicates that the vertex of the parabola represents the maximum point.

The x-coordinate of the vertex can be found using the formula:

[tex]x = \frac{-b}{2a}[/tex]

In our case, a = -3 and b = 12. Plugging these values into the formula, we get:

[tex]x = \frac{-12}{2 \cdot (-3)}\\\\x = \frac{-12}{-6}\\\\x = 2[/tex]

To find the maximum profit, we substitute the x-coordinate of the vertex into the profit function:

[tex]Z = -3(2)^2 + 12(2) + 25\\\\Z = -12 + 24 + 25\\\\Z = 37[/tex]

Therefore, the maximum profit is 37.

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(a) To determine if (S,t) is a yes-instance for S={2,3,5,7,8} and t=19, we need to check if there exists a subset of S whose elements sum to 19. In this case, we can choose the subset T={2,7,10} where the elements sum to 19. Thus, (S,t) is a yes-instance.

(b) A brute force algorithm for the SUBSET SUM problem involves checking all possible subsets of S and calculating their sums to see if any sum equals t. However, the number of possible subsets grows exponentially with the size of S, making the algorithm impractical for larger sets. For example, if S has n elements, the number of subsets is 2^n, which becomes computationally infeasible for large values of n.
(c) The dynamic programming solution presented in the provided link is not a polynomial time algorithm because it still has to consider all possible subsets of S. Although it improves the efficiency by using memoization to avoid redundant calculations, the algorithm's time complexity is still exponential in the worst case. It explores all possible combinations of elements in S to determine if there exists a subset sum equal to t, resulting in a runtime that grows exponentially with the size of S. Thus, it cannot be classified as a polynomial time algorithm.

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The critical value for a 95% confidence interval estimate of the population proportion of managers who have caught salespeople cheating on an expense report is \boxed{1.96}.

To determine the critical value for a 95% confidence interval estimate of the population proportion of managers who have caught salespeople cheating on an expense report, we can use the z-score formula.

The formula for z-score is given by: z = \frac{\hat{p} - p}{\sqrt{\frac{pq}{n}}}

where; \hat{p} is the sample proportion p is the population proportion q is 1 - population proportion n is the sample size

We need to find the critical value for a 95% confidence interval.

The critical value is denoted by z_{\alpha/2} and can be obtained from the standard normal distribution table.

For a 95% confidence interval, \alpha = 1 - 0.95 = 0.05.

Therefore, \alpha/2 = 0.025 and z_{\alpha/2} = 1.96.

Hence, the critical value for a 95% confidence interval estimate of the population proportion of managers who have caught salespeople cheating on an expense report is \boxed{1.96}.

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The indefinite integral of x^2 / (x - 5) dx is x + 5 ln|x - 5| + c.

To find the indefinite integral of x^2 / (x - 5) dx, we can use the method of partial fractions.

First, we need to decompose the fraction:

To find the values of A and B, we can multiply both sides by (x - 5) and equate the coefficients of like terms:

Expanding and collecting like terms:

Now, we can equate the coefficients of x^2, x, and the constant term separately:

Solving these equations, we find A = 1 and B = 5.

Now, we can rewrite the integral as:

Integrating each term separately:

The integral of 1 with respect to x is simply x, and the integral of (5 / (x - 5)) dx can be found by substituting u = x - 5, which gives us du = dx:

Therefore, the indefinite integral of x^2 / (x - 5) dx is:

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The mean of the given distribution will be A.  4.45.

To find the mean of a distribution, you can follow these steps:

Mean (μ) = ∑(X * P(X)).

Using the provided distribution: X: 1 4 6 8

P(X): 0.39 0.18 0.05 0.38

Calculating the mean: Mean (μ) = (1 * 0.39) + (4 * 0.18) + (6 * 0.05) + (8 * 0.38)

= 0.39 + 0.72 + 0.3 + 3.04

= 4.45

Therefore, the mean of the given distribution is 4.45.

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The weight that separates the heaviest 15% from the rest is approximately 1075.08 pounds.

Find the probability that a randomly selected yearling Angus steer weighs more than 1300 pounds.

We are given that the mean weight is 1150 pounds and the standard deviation is 80 pounds.

The z-score of 1300 is `(1300-1150)/80 = 1.875`. We can find the probability using the z-table.

Z(>1.875) = 1 - P(Z<1.875)

From the z-table, P(Z<1.87) = 0.9693

So, P(Z>1.875) = 1 - 0.9693

= 0.0307

Find the probability that a randomly selected yearling Angus steer weighs between 1000 and 1200 pounds.

We are given that the mean weight is 1150 pounds and the standard deviation is 80 pounds. We can find the z-score of 1000 and 1200 as follows:

z1 = (1000-1150)/80

= -1.875z2

= (1200-1150)/80

= 0.625

We can find the probability using the z-table.

P(1000 < X < 1200) = P(-1.875 < Z < 0.625)

= P(Z < 0.625) - P(Z < -1.875)

From the z-table, P(Z < 0.625) = 0.7357 and P(Z < -1.875) = 0.0307.

So, P(1000 < X < 1200) = 0.7357 - 0.0307 = 0.7050Part 3 of 3: Find the weight that separates the heaviest 15% from the rest.

We can find the z-score using the z-table:P(X > x) = 0.15 => P(Z > z) = 0.15

From the z-table, the z-score corresponding to 0.15 is -1.0364.-1.0364 = (x - 1150)/80=> x = -1.0364*80 + 1150=> x = 1075.08

Therefore, the weight that separates the heaviest 15% from the rest is approximately 1075.08 pounds.

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The answer is option A. No, there is a good chance that 12 randomly selected adult male passengers will exceed the elevator capacity.

Probability that it is overloaded if 12 adult male passengers have a mean weight greater than 157 lb is 0.0229.Round to four decimal places as needed.Based on the calculations the elevator does not appear to be safe.The solution for the given problem is as follows:

Given that, the maximum capacity of the elevator is 1884 lb - 12 passengers.

We can write as below:

Maximum capacity per person=1884/12=157lb.

And, weights of males are normally distributed with a mean of 165 lb and a standard deviation of 32 lb.Thus, Z = (157-165) / (32 / √12) = -1.7321Then, P(Z > -1.7321) = 0.9586

Hence, the probability that it is overloaded if 12 adult male passengers have a mean weight greater than 157 lb is:P(Z > -1.7321) = 1 - P(Z < -1.7321) = 1 - 0.0229 = 0.9771 (rounded off to 4 decimal places).This probability is greater than 5% and therefore, the elevator does not appear to be safe.

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The expected value of the die roll is 2.47.  Finding the value of kProbability is the measure of the likelihood of an event taking place. The sum of the probability of all events occurring must equal one, otherwise, the set of events would be incomplete, which is not possible.

Therefore, we have 0.28 + k + 0.15 + 0.3 = 1 where k is the probability of getting 2 on the die.Solving for k:k = 1 - 0.28 - 0.15 - 0.3k = 0.27Therefore, the value of k is 0.27. b. Calculating the expected valueThe expected value of the die roll is the sum of each score multiplied by its probability of occurrence. This is also called the mean of the probability distribution, given by: E(X) = ΣxP(x)where X is the random variable and P(x) is the probability of X being equal to x.Using the given table of probabilities:E(X) = 1(0.28) + 2(0.27) + 3(0.15) + 4(0.3)E(X) = 0.28 + 0.54 + 0.45 + 1.2E(X) = 2.47Therefore, the expected value of the die roll is 2.47.

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The graph of a tangent function in the form y = atan(bx − c) + d has a period of pi/|b|. When b = 1/2, the period is pi. This means that the graph will repeat itself every pi units on the x-axis. The correct option is the second graph.

The graph of the tangent function in the first option has a period of 2pi. This is not consistent with the period of a tangent function with b = 1/2.

The graph of the tangent function in the second option has a period of pi. This is consistent with the period of a tangent function with b = 1/2.

The graph of the tangent function in the third option does not have a period of pi. This is not consistent with the period of a tangent function with b = 1/2.

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The equation that represents the amount of money Adam will owe in one year, without making any payments, is x = $5,600 + ($5,600 * 0.042).

To calculate the amount of money Adam will owe in one year, we need to consider the initial principal amount borrowed and the simple interest charged by the bank.

The bank charges 4.2% simple interest each year on the borrowed amount.

The formula for calculating simple interest is:

                         Interest = Principal * Rate * Time

In this case, the principal amount borrowed is $5,600 and the rate is 4.2% (or 0.042 in decimal form). Since we are calculating the amount owed in one year, the time is 1.

Plugging these values into the formula, we get:

Interest = $5,600 * 0.042 * 1

Simplifying the equation, we have:

Interest = $5,600 * 0.042

Therefore, the equation representing the amount of money Adam will owe in one year, without making any payments, is x = $5,600 + ($5,600 * 0.042). This equation calculates the principal amount plus the interest accrued in one year.

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The line integral of xy^2 ds along the right half of the circle x^2 + y^2 = 25, oriented counterclockwise, is 0.

To evaluate the line integral, we first parameterize the curve C, which is the right half of the circle x^2 + y^2 = 25. In polar coordinates, the equation of the circle can be written as r = 5, and the right half of the circle corresponds to the range 0 ≤ θ ≤ π.

Let's express the curve C in terms of the parameter θ:

x = 5cosθ

y = 5sinθ

Next, we need to find the differential arc length ds. In polar coordinates, the differential arc length is given by ds = r dθ. Substituting r = 5, we have ds = 5dθ.

Now, let's rewrite the line integral in terms of the parameter θ:

∫C xy^2 ds = ∫(0 to π) (5cosθ)(5sinθ)^2 (5dθ)

Simplifying the integrand:

∫(0 to π) 125cosθsin^2θ dθ

Since sin^2θ = 1/2 - (1/2)cos2θ, we can rewrite the integral as:

∫(0 to π) 125cosθ(1/2 - (1/2)cos2θ) dθ

Expanding and simplifying:

∫(0 to π) (125/2)cosθ - (125/2)cosθcos2θ dθ

The integral of cosθ with respect to θ is sinθ, and the integral of cosθcos2θ with respect to θ is (1/3)sin3θ. Therefore, the line integral becomes:

(125/2)sinθ - (125/6)sin3θ evaluated from 0 to π.

Substituting the limits:

[(125/2)sinπ - (125/6)sin3π] - [(125/2)sin0 - (125/6)sin30]

Since sinπ = 0 and sin0 = 0, the line integral simplifies to:

0 - [(125/6)(1/2)]

= -125/12

Therefore, the line integral of xy^2 ds along the right half of the circle x^2 + y^2 = 25, oriented counterclockwise, is -125/12.

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The concept of consecutive integers is explained as follows:

Consecutive numbers, or consecutive integers, are integers that follow each other in order. The difference between any two consecutive numbers is always 1. For example, the consecutive numbers starting from 1 would be 1, 2, 3, 4, 5, and so on. Similarly, the consecutive numbers starting from -3 would be -3, -2, -1, 0, 1, 2, and so on.

It is important to note that if we have a consecutive sequence of integers, one number will be even, and the next number will be odd. This is because the parity (evenness or oddness) alternates as we move through consecutive integers.

Furthermore, the sum of two consecutive numbers (and, in fact, the sum of any number of consecutive numbers) is always an odd number. This is because when we add an even number to an odd number, the result is always an odd number.

To generate a sequence of consecutive integers, we can start with any integer n and then use n, n+1, n+2, and so on to obtain consecutive integers. For example, if n is an integer, then n, n+1, and n+2 will be consecutive integers.

Here are some examples of consecutive integers:

- Starting from 1: 1, 2, 3, 4, 5, ...

- Starting from -3: -3, -2, -1, 0, 1, 2, ...

- Starting from 1004: 1004, 1005, 1006, 1007, ...

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10% of the students at this university are foreign.

Let's assume the total number of college students at this university is represented by 'T'.

We know that 6% of all college students are foreign students who smoke. Therefore, the number of foreign students who smoke is 0.06T.

We also know that 60% of all foreign college students smoke.

So, the number of foreign students at the university is:

= (0.06T) / 0.6

= 0.1T.

To find the percentage of foreign students, we divide the number of foreign students (0.1T) by the total number of students (T) and multiply by 100 which gives:

= (0.1T / T) * 100

= 10%.

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Percentage of data more than one standard deviation from the mean = 32%

To obtain the percentage of data that is more than one standard deviation from the mean, we can use the properties of the normal distribution.

For a normal distribution with a mean (µ) of 100 and a standard deviation (σ) of 15, we know that approximately:

- 68% of the data falls within one standard deviation of the mean.

- 95% of the data falls within two standard deviations of the mean.

- 99.7% of the data falls within three standard deviations of the mean.

Therefore, to find the percentage of data that is more than one standard deviation from the mean, we subtract the percentage within one standard deviation from 100%.

= 100% - 68%

= 32%

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The 33rd percentile of the sample. The answer is 70.

We have a sample of 8 scores: (51, 93, 93, 80, 70, 76, 64, 79).

The first step to finding the 33rd percentile is to put the data in order.

This gives us: (51, 64, 70, 76, 79, 80, 93, 93).Next, we calculate the rank of the 33rd percentile.

To do this, we use the following formula:

Rank = (percentile/100) x n

where percentile = 33, n = 8Rank = (33/100) x 8 = 2.64 (rounded to 3)

Therefore, the 33rd percentile is the score that is ranked 3rd.

From the ordered data, we see that the score ranked 3rd is 70.

Hence, the answer is 70

To determine the 33rd percentile of a sample of 8 scores (51, 93, 93, 80, 70, 76, 64, 79), we use the formula Rank = (percentile/100) x n. The score ranked 3rd in the ordered data is 70, which is therefore the 33rd percentile of the sample. The answer is 70.

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The following statements are true: Shipping cost is 12.99 if the customer's order is less than or equal to $40.

Free shipping if the customer's order is $1000.By spending more than $500, customers save on shipping costs by paying the least.

The table rate module calculates the shipping rate depending on the order's destination and cart weight.

The shipping rate is calculated using the shipping table and the calculation type used in the module.

Therefore, the following statements are true:

Shipping cost is 12.99 if the customer's order is less than or equal to $40.

If the customer's order is less than or equal to $40, the shipping cost will be 12.99, as per the given shipping table.

Shipping cost is 9.99 if the customer's order is $50.

If the customer's order is $50, the shipping cost will be 9.99, as per the given shipping table.

Shipping cost is free if the customer's order is $1000.

If the customer's order is $1000, the shipping cost will be free, as per the given shipping table.

By spending more than $500, customers save on shipping costs by paying the least.

If the customer spends more than $500, they save on shipping costs by paying the least, as per the given shipping table.

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Hence, X has an exponential distribution with parameter λ = i!. (a) If X₁, X₂,..., X₁ are independent observations for XIf X1, X2,..., Xn are independent and identically distributed random variables, then the sample mean[tex]$$\overline X = \frac{1}{n}\sum_{i=1}^{n} X_i$$[/tex]

also follows the exponential distribution with the same parameter λ.

Given that, X is a continuous random variable with E(X)=i! for i=0,1,2,...

(a) Show that X has an exponential distribution. State its parameter.A random variable X is said to follow the exponential distribution if its probability density function is given by;

[tex]$$ f(x)[/tex] =

[tex]\begin{cases} \lambda e^{-\lambda x} & x\ge0\\ 0 & x < 0 \end{cases}[/tex]

Here, X is a continuous random variable with the expectation value

[tex]$$E(X) = i!$$[/tex]

For i = 0,

E(X) = 0!

= 1

For i = 1,

E(X) = 1!

= 1

For i = 2,

E(X) = 2!

= 2

For i = 3,

E(X) = 3!

= 6

Similarly, for any i, E(X) = i!

Let us find the probability density function of X.

[tex]f(x) = \frac{dF(x)}{dx}[/tex]

Here, F(x) is the cumulative distribution function of X. We have,

[tex]$$F(x) = P(X\le x)$$$$[/tex]

=[tex]\int_{-\infty}^{x} f(t) dt$$$$[/tex]

=[tex]\int_{0}^{x} f(t) dt$$[/tex]

As f(x) = 0 for x<0, the lower limit of the integral can be taken as 0.

[tex]$$F(x) = \int_{0}^{x} f(t) dt$$$$[/tex]

=[tex]\int_{0}^{x} \lambda e^{-\lambda t} dt$$$$[/tex]

=[tex]\left[ -e^{-\lambda t} \right]_{0}^{x}$$$$[/tex]

=[tex]1-e^{-\lambda x}$$[/tex]

Now, let us differentiate F(x) with respect to x.

[tex]$$f(x) = \frac{dF(x)}{dx}$$$$[/tex]

=[tex]\frac{d}{dx}\left(1-e^{-\lambda x}\right)$$$$[/tex]

= [tex]\lambda e^{-\lambda x}$$[/tex]

Comparing this equation with the standard pdf of the exponential distribution, we have;

[tex]$$\lambda=\\[/tex]

[tex]E(X) = i!$$[/tex]

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To find an explicit formula for the nth term of the sequence satisfying[tex]\(a_1 = 0\) and \(a_n = 2a_{n-1} + 1\) for \(n \geq 2\),[/tex]  we can use recursive formula to generate the terms of the sequence.

Given:

[tex]\(a_1 = 0\)\\\(a_n = 2a_{n-1} + 1\) for \(n \geq 2\)[/tex]

Using the recursive formula, we can generate the terms of the sequence as follows:

[tex]\(a_2 = 2a_1 + 1 = 2(0) + 1 = 1\)\\\(a_3 = 2a_2 + 1 = 2(1) + 1 = 3\)[/tex]

[tex]\(a_4 = 2a_3 + 1 = 2(3) + 1 = 7\)\\\(a_5 = 2a_4 + 1 = 2(7) + 1 = 15\)[/tex]

From the pattern, we observe that the nth term of the sequence is given by [tex]\(2^{n-2} - 1\).[/tex]

Therefore, the explicit formula for the nth term of the sequence satisfying [tex]\(a_1 = 0\) and \(a_n = 2a_{n-1} + 1\) for \(n \geq 2\) is: \\\\\a_n = 2^{n-2} - 1.\][/tex]

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The domain of the function g(x) = x² - 100 is(-∞, -10] ∪ [10, ∞)

The domain of the function g(x) = x² - 100 can be expressed using interval notation as follows:Domain: (-∞, -10] ∪ [10, ∞)

The given function g(x) = x² - 100 is a polynomial function.

There are no restrictions or limitations on the domain of a polynomial function, i.e., all real numbers can be used as input to the function.

However, in this particular function, we have a subtraction of 100.

Since we cannot have a square root of a negative number, we need to ensure that the radicand (x² - 100) is non-negative.

Thus, we have:x² - 100 ≥ 0⇒ x² ≥ 100⇒ x ≤ -10 or x ≥ 10

So, the domain of the function g(x) = x² - 100 is(-∞, -10] ∪ [10, ∞)

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The length of the spiraling polar curve from θ = 0 to θ = 2π is:

8/3 √ (1 + ln²⁸) [e^(12π) - 1]

To find the length of the spiraling polar curve, we will use the arc length formula.

L = ∫baf(θ)√ (r² + [f'(θ)]²).dθ

where a and b are the initial and final values of θ, and f(θ) is the polar equation.

The polar curve given is:

r = 8e^(6θ).

The length of the spiraling polar curve can be calculated as follows:

L = ∫02πf(θ) √ (r² + [f'(θ)]²).dθ.

Let us now find f(θ) and f'(θ). Since:

r = 8e^(6θ),

we know that:

r = f(θ) and f'(θ) = 8e^(6θ)*ln8.

f(θ) = r = 8e^(6θ).

f'(θ) = 8e^(6θ)*ln8.

Therefore, substituting these values, we obtain:

L = ∫02π 8e^(6θ)√(64e^(12θ) + 64e^(12θ)ln²⁸)dθ

L= 8∫02πe^(6θ)√(1 + ln²⁸)dθ

L= 8 √ (1 + ln²⁸) ∫02πe^(6θ)dθ

L= 8 √ (1 + ln²⁸) [e^(6θ)/6] 02π

L= 8/3 √ (1 + ln²⁸) [e^(12π) - 1].

Therefore, the length of the spiraling polar curve from θ = 0 to θ = 2π is 8/3 √ (1 + ln²⁸) [e^(12π) - 1].

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The value of p is 0.2525. A radio station conducts a random phone survey of 332 people. The survey results showed that 70 people have heard the ad and recognize the company's product. The company will only renew its advertising contract with the local radio station if more than 33% of the city's residents have heard the ad and recognize the company's product.

A 5% level of significance is used for this test. The null hypothesis states that the proportion of residents who have heard the ad and recognize the company's product is less than or equal to 0.33. The alternative hypothesis, on the other hand, states that the proportion of residents who have heard the ad and recognize the company's product is more than 0.33.  Calculating the value of p using the given data:

Formula: Z = (p - P) / √((P(1 - P)) / n)

Here,

Sample proportion (P) = 70/332

= 0.2108

Sample size (n) = 332

Level of significance = 0.05 (As given in the question, the significance level is 5%)

Z score corresponding to a 5% significance level

   = 1.645√((P(1 - P)) / n)

= √((0.2108(1 - 0.2108)) / 332)

= 0.0253P = Z √((P(1 - P)) / n) + P

= 1.645 × 0.0253 + 0.2108

= 0.2525

The objective of this problem is to determine the proportion of residents who have heard the ad and recognize the company's product so that the company can renew its advertising contract with the local radio station. The radio station takes a sample of 332 individuals to determine the value of p. Of those 332 individuals, 70 have heard the ad and recognize the company's product.

A 5% significance level is used for this test. The null hypothesis states that the proportion of residents who have heard the ad and recognize the company's product is less than or equal to 0.33. The alternative hypothesis, on the other hand, states that the proportion of residents who have heard the ad and recognize the company's product is more than 0.33. If the null hypothesis is rejected, the company will renew its contract with the radio station.

The Z-score test statistic is used to determine the value of p.

The formula for the Z-score is

Z = (p - P) / √((P(1 - P)) / n).

Here, P is the sample proportion of individuals who have heard the ad and recognize the company's product, n is the sample size, and p is the proportion of residents who have heard the ad and recognize the company's product that the company needs to renew its contract with the radio station. To calculate the value of p, first, we need to find the Z score corresponding to a 5% significance level. A Z score of 1.645 corresponds to a 5% significance level.

The formula for p is P = Z √((P(1 - P)) / n) + P.

We know the values of P and n, and we have calculated the value of Z. We can now substitute these values in the formula to obtain the value of p. The value of p is 0.2525. Since p is greater than 0.33, the null hypothesis is rejected, and the company can renew its advertising contract with the radio station.

The proportion of residents who have heard the ad and recognize the company's product is more than 0.33, according to the sample data. As a result, the radio station has demonstrated that it has fulfilled the company's requirements, and the company can renew its contract with the radio station.

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a. To construct a 95% confidence interval for B, we can use the estimated coefficient and its standard error. The formula for the confidence interval is:

B ± t * SE(B)

Where B is the estimated coefficient, SE(B) is the standard error of the coefficient, and t is the critical value from the t-distribution based on the desired confidence level and the degrees of freedom (n - 2).

b. Hypothesis test:

(i) The null hypothesis (H0): B = 20 (there is no effect of smoking on birth weight).

The alternative hypothesis (Ha): B ≠ 20 (there is an effect of smoking on birth weight).

(ii) The test statistic is calculated by dividing the estimated coefficient by its standard error:

t = (B - hypothesized value) / SE(B)

(iii) The p-value is the probability of observing a test statistic as extreme or more extreme than the calculated value, assuming the null hypothesis is true.

(iv) The level of significance is the predetermined threshold for rejecting the null hypothesis. Common levels are 0.05 and 0.01.

(v) The test is two-sided because the alternative hypothesis allows for both positive and negative effects of smoking on birth weight.

(vi) Based on the calculated test statistic and p-value, we can compare the p-value to the level of significance. If the p-value is less than the level of significance, we reject the null hypothesis; otherwise, we fail to reject the null hypothesis.

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