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HW2 ISDS 361B

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Where applicable, round your answers to three decimal places, e.g. 0.657.

1. A normally distributed population has a mean of 40 and a standard deviation of 12. What does the

central limit theorem say about the sampling distribution of the mean if samples of size 100 are

drawn from this population?

2. A sample of n = 16 observations is drawn from a normal population with µ = 1,000 and σ = 200. Find

the following.

a. �(�ത > 1,050)

b. �(�ത < 960)

c. �(�ത > 1,100)

3. The heights of North American women are normally distributed with a mean of 64 inches and a

standard deviation of 2 inches.

a. What is the probability that a randomly selected woman is taller than 66 inches?

b. A random sample of four women is selected. What is the probability that the sample mean

height is greater than 66 inches?

c. What is the probability that the mean height of a random sample of 100 women is greater

than 66 inches?

4. The number of pizzas consumed per month by university students is normally distributed with a

mean of 10 and a standard deviation of 3.

a. What proportion of students consumes more than 12 pizzas per month?

b. What is the probability that in a random sample of 25 students more than 275 pizzas are

consumed? (Hint: What is the mean number of pizzas consumed by the sample of 25

students?)

Use the normal approximation without the correction factor to find the probabilities in the exercises 5, 6,

and 7.

5.

a. In a binomial experiment with n = 300 and p = .5, find the probability that � is greater than

60%.

b. Repeat part (a) with p = .55

c. Repeat part (a) with p = .6

6. A commercial for a manufacturer of household appliances claims that 3% of all its products require a

service call in the first year. A consumer protection association wants to check the claim by

surveying 400 households that recently purchased one of the company’s appliances. What is the

probability that more than 5% require a service call within the first year? What would you say about

the commercial’s honesty if in a random sample of 400 households 5% report at least one service

call? HW2 ISDS 361B

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7. An accounting professor claims that no more than one-quarter of undergraduate business student

will major in accounting. What is the probability that in a random sample of 1,200 undergraduate

business students, 336 or more will major in accounting?

8. Suppose that we have two normal populations with the means and standard deviations listed here.

If random samples of size 25 are drawn from each population, what is the probability that the mean

of sample 1 is greater than the mean of sample 2?

Population 1: µ = 40, σ = 6

Population 2: µ = 38, σ = 8

9. A factory’s worker productivity is normally distributed. One worker produces an average of 75 units

per day with a standard deviation of 20. Another worker produces at an average rate of 65 per day

with a standard deviation of 21. What is the probability that during one week (5 working days),

worker 1 will outproduce worker 2?

10. The average North American loses an average of 15 days per year to colds and flu. The natural

remedy echinacea reputedly boosts the immune system. One manufacturer of echinacea pills claims

that consumers of its product will reduce the number of days lost to colds and flu by one-third. To

test the claim, a random sample of 50 people was drawn. Half took echinacea, and the other half

took placebos. If we assume that the standard deviation of the number of days lost to colds and flu

with and without echinacea is 3 days, find the probability that the mean number of days lost for

echinacea users is less than that for nonusers.

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