Posted: December 8th, 2022

**Problem 1**

The next state lottery will have the following payoffs possible with their associated probabilities.

Payoff Probability

$2.00 0.0500

$25.00 0.0100

$100.00 0.0050

$500.00 0.0010

$5,000.00 0.0005

$10,000.00 0.0001

You buy a single ticket.

a. Referring to the problem statement, the probability that you win any money is ________.

b. Referring to the problem statement, the probability that you win at least $100.00 is ________.

c. Referring to the problem statement, if you have a winning ticket, the probability that you win at least $100.00 is ________.

d. Referring to the problem statement, if the ticket you purchased cost $10.00, what is the expected value of your winning (loss)?

**Problem 12**

A certain type of new business succeeds 60% of the time. Suppose that 3 such businesses open (where they do not compete with each other, so it is reasonable to believe that their relative successes would be independent).

a. Referring to the problem statement, the probability that all 3 businesses succeed is ________.

b. Referring to the problem statement, the probability that all 3 businesses fail is ________.

c. Referring to the problem statement, the probability that at least 1 business succeeds is ________.

d. Referring to the problem statement, the probability that exactly 1 business succeeds is ________.

**Problem 13**

The following table contains the probability distribution for *X *= the number of retransmissions necessary to successfully transmit a 1024K data package through a double satellite media.

*X *0* *1* *2* *3

*P*(*X*) 0.35 0.35 0.25 0.05

a. Referring to the problem statement, the probability of no retransmissions is ________.

b. Referring to the problem statement, the probability of at least one retransmission is ________.

c. Referring to the problem statement, the mean or expected value for the number of retransmissions is ________.

**14.** The number of power outages at a nuclear power plant has a Poisson distribution with a mean of 6 outages per year. The probability that there will be exactly 3 power outages in a year is ____________.

**15.** For some positive value of *Z*, the probability that a standard normal variable is between 0 and *Z *is 0.3770. The value of *Z *is:

a. 0.18

b. 0.81

c. 1.16

d. 1.47

**16.** Given that *X *is a normally distributed random variable with a mean of 50 and a standard deviation of 2, find the probability that *X *is between 47 and 54.

**17. **A food processor packages orange juice in small jars. The weights of the filled jars are approximately normally distributed with a mean of 10.5 ounces and a standard deviation of 0.3 ounce. Find the proportion of all jars packaged by this process that have weights that fall below 10.875 ounces.

**19.** The true length of boards cut at a mill with a listed length of 10 feet is normally distributed with a mean of 123 inches and a standard deviation of 1 inch. What percentage of the boards will be over 125 inches in length?

**20.** The amount of time necessary for assembly line workers to complete a product is a normal random variable with a mean of 15 minutes and a standard deviation of 2 minutes. So, 90% of the products require more than __________ minutes for assembly.

**21.** Patients arriving at an outpatient clinic follow an exponential distribution at a rate of 15 patients per hour. What is the probability that the next arrival will be more than 5 minutes?

**34.** The amount of time required for an oil and filter change on an automobile is normally distributed with a mean of 45 minutes and a standard deviation of 10 minutes. A random sample of 16 cars is selected. What is the probability that the sample mean will be between 39 and 48 minutes?

**36.** If you were constructing a 99% confidence interval of the population mean based on a sample of *n *= 25 where the standard deviation of the sample *s *= 0.05, the critical value of *t *will be:

a. 2.7969

b. 2.7874

c. 2.4922

d. 2.4851

**37.** It is desired to estimate the average total compensation of CEOs in the Service industry. Data were randomly collected from 18 CEOs and the 97% confidence interval was calculated to be ($2,181,260, $5,836,180). Which of the following interpretations is correct?

a. 97% of the sampled total compensation values fell between $2,181,260 and $5,836,180.

b. We are 97% confident that the mean of the sampled CEOs falls in the interval $2,181,260 to $5,836,180.

c. In the population of Service industry CEOs, 97% of them will have total compensations that fall in the interval $2,181,260 to $5,836,180.

d. We are 97% confident that the average total compensation of all CEOs in the Service industry falls in the interval $2,181,260 to $5,836,180.

38. A confidence interval was used to estimate the proportion of statistics students that are females. A random sample of 72 statistics students generated the following 90% confidence interval: (0.438, 0.642). Based on the interval above, is the population proportion of females equal to 0.60?

a. No, and we are 90% sure of it.

b. No. The proportion is 54.17%.

c. Maybe. 0.60 is a believable value of the population proportion based on the information above.

d. Yes, and we are 90% sure of it.

**39.** As an aid to the establishment of personnel requirements, the director of a hospital wishes to estimate the mean number of people who are admitted to the emergency room during a 24-hour period. The director randomly selects 64 different 24-hour periods and determines the number of admissions for each. For this sample, X-bar = 19.8 and *s*2 = 25. Estimate the mean number of admissions per 24-hour period with a 95% confidence interval.

**Problem 44**

A quality control engineer is interested in the mean length of sheet insulation being cut automatically by machine. The desired length of the insulation is 12 feet. It is known that the standard deviation in the cutting length is 0.15 feet. A sample of 70 cut sheets yields a mean length of 12.14 feet. This sample will be used to obtain a 99% confidence interval for the mean length cut by machine.

a. Referring to the problem statement, the confidence interval goes from ________ to ________.

b. Referring to the problem statement, the confidence interval indicates that the machine is not working properly.

a. True

b. False

c. Referring to the problem statement, suppose the engineer had decided to estimate the mean length to within 0.03 with 99% confidence. Then the sample size would be ________.

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