# stats and econ assignment

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5.21 Suppose that *x* is a binomial random variable with , , and .

a. Write the binomial formula for this situation and list the possible values of *x*.

b. For each value of *x*, calculate *p*(*x*).

c. Find .

d. Find .

e. Find .

f. Find .

g. Find .

h. Use the probabilities you computed in part *b* to calculate the mean, *μx*, the variance, *σ*2*x*, and the standard deviation, *σx*, of this binomial distribution. Show that the formulas for *μx*, *σ*2*x*, and *σx* given in this section give the same results.

i. Calculate the interval . Use the probabilities of part *b* to find the probability that *x* will be in this interval.

5.22 Thirty percent of all customers who enter a store will make a purchase. Suppose that six customers enter the store and that these customers make independent purchase decisions.

a. Let *x* = the number of the six customers who will make a purchase. Write the binomial formula for this situation.

b. Use the binomial formula to calculate

0. The probability that exactly five customers make a purchase.

0. The probability that at least three customers make a purchase.

0. The probability that two or fewer customers make a purchase.

0. The probability that at least one customer makes a purchase.

5.27 The January 1986 mission of the Space Shuttle *Challenger* was the 25th such shuttle mission. It was unsuccessful due to an explosion caused by an O-ring seal failure.

a. According to NASA, the probability of such a failure in a single mission was 1/60,000. Using this value of *p* and assuming all missions are independent, calculate the probability of no mission failures in 25 attempts. Then calculate the probability of at least one mission failure in 25 attempts.

b. According to a study conducted for the Air Force, the probability of such a failure in a single mission was 1/35. Recalculate the probability of no mission failures in 25 attempts and the probability of at least one mission failure in 25 attempts.

c. Based on your answers to parts *a* and *b*, which value of *p* seems more likely to be true? Explain.

d. How small must *p* be made in order to ensure that the probability of no mission failures in 25 attempts is .999?

**CHAPTER 6—Continuous Random Variables**

6.6 Suppose that the random variable *x* has a uniform distribution with *c* = 2 and *d* = 8.

a. Write the formula for the probability curve of *x*, and write an interval that gives the possible values of *x*.

b. Graph the probability curve of *x*.

c. Find *P *(3 ≤ *x* ≤ 5).

d. Find *P *(1.5 ≤ *x* ≤ 6.5).

e. Calculate the mean μ*x*, variance σ2*x*, and standard deviation σ*x*.

f. Calculate the interval [μ ± 2σ*x*]. What is the probability that *x* will be in this interval?

6.8 Assume that the waiting time *x* for an elevator is uniformly distributed between zero and six minutes.

a. Write the formula for the probability curve of *x*.

b. Graph the probability curve of *x*.

c. Find *P *(2 ≤ *x* ≤ 4).

d. Find *P *(3 ≤ *x* ≤ 6).

e. Find *P *({0 ≤ *x* ≤ 2} or {5 ≤ *x* ≤ 6}).

6.9 Refer to Exercise 6.8.

a. Calculate the mean, μ*x*, the variance, σ2*x*, and the standard deviation, σ*x*.

b. Find the probability that the waiting time of a randomly selected patron will be within one standard deviation of the mean.

6.19 Let *x* be a normally distributed random variable having mean *μ* = 30 and standard deviation *σ* = 5. Find the *z* value for each of the following observed values of *x*:

a. *x* = 25

b. *x* = 15

c. *x* = 30

d. *x* = 40

e. *x* = 50

In each case, explain what the *z* value tells us about how the observed value of *x* compares to the mean, *μ*.

6.20 If the random variable *z* has a standard normal distribution, sketch and find each of the following probabilities:

a. *P *(0 ≤ *z* ≤ 1.5)

b. *P *(*z* ≥ 2)

c. *P *(*z* ≤ 1.5)

d. *P *(*z* ≥ −1)

e. *P *(*z* ≤ −3)

f. *P *(−1 ≤ *z* ≤ 1)

g. *P *(−2.5 ≤ *z* ≤ .5)

h. *P *(1.5 ≤ *z* ≤ 2)

i. *P *(−2 ≤ *z* ≤ −.5)

6.21 Suppose that the random variable *z* has a standard normal distribution. Sketch each of the following *z* points and use the normal table to find each *z* point.

a. *z*.01

b. *z*.05

c. *z*.02

d. − *z*.01

e.− *z*.05

f.− *z*.10

***6.22** Suppose that the random variable *x* is normally distributed with mean *μ* = 1, 000 and standard deviation *σ* = 100. Sketch and find each of the following probabilities:

**a. ***P *(1, 000 ≤ *x* ≤ 1, 200)

**b. ***P *(*x* > 1, 257)

**c. ***P *(*x* < 1, 035)

**d. ***P *(857 ≤ *x* ≤ 1, 183)

**e. ***P *(*x* ≤ 700)

**f. ***P *(812 ≤ *x* ≤ 913)

**g. ***P *(*x* > 891)

**h. ***P *(1, 050 ≤ *x* ≤ 1, 250)

***6.23** Suppose that the random variable *x* is normally distributed with mean *μ* = 500 and standard deviation *σ* = 100. For each of the following, use the normal table to find the needed value *k*. In each case, draw a sketch.

**a. ***P *(*x* ≥ *k*) = .025

**b. ***P *(*x* ≥ *k*) = .05

**c. ***P *(*x* < *k*) = .025

**d. ***P *(*x* ≤ *k*) = .015

**e. ***P *(*x* < *k*) = .985

**f. ***P *(*x* > *k*) = .95

**g. ***P *(*x* ≤ *k*) = .975

**h. ***P *(*x* ≥ *k*) = .0228

**i. ***P *(*x* > *k*) = .9772

6.25 Weekly demand at a grocery store for a brand of breakfast cereal is normally distributed with a mean of 800 boxes and a standard deviation of 75 boxes.

4. What is the probability that weekly demand is

0. 959 boxes or less?

0. More than 1, 004 boxes?

0. Less than 650 boxes or greater than 950 boxes?

4. The store orders cereal from a distributor weekly. How many boxes should the store order for a week to have only a 2.5 percent chance of running short of this brand of cereal during the week?

6.32 Two students take a college entrance exam known to have a normal distribution of scores. The students receive raw scores of 63 and 93, which correspond to *z* scores (often called the standardized scores) of −1 and 1.5, respectively. Find the mean and standard deviation of the distribution of raw exam scores.

***6.39** Suppose that *x* has a binomial distribution with *n* = 200 and *p* = .4.

a. Show that the normal approximation to the binomial can appropriately be used to calculate probabilities about *x*.

b. Make continuity corrections for each of the following, and then use the normal approximation to the binomial to find each probability:

1. *P *(*x* = 80)

1. *P *(*x* ≤ 95)

1. *P *(*x* < 65)

1. *P *(*x* ≥ 100)

1. *P *(*x* > 100)

***6.40** Repeat Exercise 6.39 with *n* = 200 and *p* = .5.

6.51 The length of a particular telemarketing phone call, *x*, has an exponential distribution with mean equal to 1.5 minutes.

a. Write the formula for the exponential probability curve of *x*.

b. Sketch the probability curve of *x*.

c. Find the probability that the length of a randomly selected call will be:

1. No more than three minutes.

1. Between one and two minutes.

1. More than four minutes.

1. Less than 30 seconds.

6.53 Suppose that the number of accidents occurring in an industrial plant is described by a Poisson distribution with an average of one accident per month. Let *x* denote the time (in months) between successive accidents.

**a. **Find the probability that the time between successive accidents is:

0. More than two months.

0. Between one and two months.

0. Less than one week (1/4 of a month).

**b. **Suppose that an accident occurs less than one week after the plant’s most recent accident. Would you consider this event unusual enough to warrant special investigation? Explain.

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