# MATHEMATICS

4.34 Ages of pennies, The histogram below shows the distribution of ages of pennies at a bank.

The mean age of the pennies is 10.44 years with a standard deviation of 9.2 years. Using the Central Limit Theorem, calculate the means and standard deviations of the distribution of the mean from random samples of size 5, 30, and 100. Comment on whether the sampling distributions shown agree with the values you compute.

4.36 Identify distributions, Part II. Four plots are presented below. The plot at the top is a distribution for a population. The mean is 60 and the standard deviation is 18. Also shown

below is a distribution of (1) a single random sample of 500 values from this population, (2) a

distribution of 500 sample means from random samples of each size 18, and (3) a distribution of

500 sample means from random samples of each size 81. Determine which plot (A, B, or C) is

which and explain your reasoning.

4.38 Stats final scores. Each year about 1500 students take the introductory statistics course at a large university. This year scores on the final exam are distributed with a median of 74 points,

a mean of 70 points, and a standard deviation of 10 points. There are no students who scored

above 100 (the maximum score attainable on the final) but a few students scored below 20 points.

(a) Is the distribution of scores on this final exam symmetric, right skewed, or left skewed?

(b) Would you expect most students to have scored above or below 70 points?

(c) Can we calculate the probability that a randomly chosen student scored above 75 using the

normal distribution?

(d) What is the probability that the average score for a random sample of 40 students is above

75?

(e) How would cutting the sample size in half affect the standard error of the mean?

4.40 CFLs. A manufacturer of compact fluorescent light bulbs advertises that the distribution of the lifespans of these light bulbs is nearly normal with a mean of 9,000 hours and a standard deviation of 1,000 hours.

(a) What is the probability that a randomly chosen light bulb lasts more than 10,500 hours?

(b) Describe the distribution of the mean lifespan of 15 light bulbs.

(c) What is the probability that the mean lifespan of 15 randomly chosen light bulbs is more than

10,500 hours?

(d) Sketch the two distributions (population and sampling) on the same scale.

(e) Could you estimate the probabilities from parts (a) and (c) if the lifespans of light bulbs had

a skewed distribution?

4.42 Spray paint. Suppose the area that can be painted using a single can of spray paint is slightly variable and follows a nearly normal distribution with a mean of 25 square feet and a standard deviation of 3 square feet.

(a) What is the probability that the area covered by a can of spray paint is more than 27 square

feet?

(b) Suppose you want to spray paint an area of 540 square feet using 20 cans of spray paint. On

average, how many square feet must each can be able to cover to spray paint all 540 square

feet?

(c) What is the probability that you can cover a 540 square feet area using 20 cans of spray paint?

(d) If the area covered by a can of spray paint had a slightly skewed distribution, could you still

calculate the probabilities in parts (a) and (c) using the normal distribution?