Question

A manufacturer produces 25-pound lifting weights. The lowest actual weight is 24 pounds, and the highest is 26 pounds. Each weight is equally likely so the distribution of weights is uniform. A sample of 100 weights is taken.

a. What is the distribution for the weights of one 25-pound lifting weight? What is the mean and standard deviation?

b. What is the distribution for the mean weight of 100 25-pound lifting weights?

c. Find the probability that the mean actual weight for the 100 weights is less than 24.9.

d. Find the probability that the mean actual weight for the 100 weights is greater than 25.2.

e. Find the 90th percentile for the mean weight for the 100 weights.

3. The length of songs in a collector’s iTunes album collection is uniformly distributed from two to 3.5 minutes. Suppose we randomly pick five albums from the collection. There are a total of 43 songs on the five albums.

a. In words, ? = _________

b. ? ~ _____________

c. In words, X ¯ = _____________

d. X ¯ ~ _____(_____,_____)

e. Find the first quartile for the average song length.

f. The IQR(interquartile range) for the average song length is from _______–_______.

4. The closing stock prices of 35 U.S. semiconductor manufacturers are given as follows. 8.625; 30.25; 27.625; 46.75; 32.875; 18.25; 5; 0.125; 2.9375; 6.875; 28.25; 24.25; 21; 1.5; 30.25; 71; 43.5; 49.25; 2.5625; 31; 16.5; 9.5; 18.5; 18; 9; 10.5; 16.625; 1.25; 18; 12.87; 7; 12.875; 2.875; 60.25; 29.25

a. In words, ? = ______________

b. i. x ¯ = _____

    ii. sx = _____

    iii. n = _____

c. Construct a histogram of the distribution of the averages. Start at x = –0.0005. Use bar widths of ten.

d. In words, describe the distribution of stock prices.

e. Randomly average five stock prices together. (Use a random number generator.) Continue averaging five pieces together until you have ten averages. List those ten averages.

f. Use the ten averages from part e to calculate the following.

i. x ¯ = _____

ii. sx = _____

5. X ~ N(60, 9). Suppose that you form random samples of 25 from this distribution. Let X ¯ be the random variable of averages.

a. Sketch the distributions of X and X ¯ on the same graph.

b. X ¯ ~ _____(_____,_____)

c. P( x ¯ < 60) = _____

d. Find the 30th percentile for the mean.

e. P(56 < x ¯ < 62) = _____

f. P(18 < x ¯ < 58) = _____

6. Suppose that the length of research papers is uniformly distributed from ten to 25 pages. We survey a class in which 55 research papers were turned in to a professor. The 55 research papers are considered a random collection of all papers. We are interested in the average length of the research papers.

a. In words, X = _____________

b. X ~ _____(_____,_____)

c. ?x = _____

d. ?x = _____

e. In words, X ¯ = ______________

f. X ¯ ~ _____(_____,_____)

g. Calculate the probability that the professor will need to read a total of more than 1,050 pages.

h. Why is it so unlikely that the average length of the papers will be less than 12 pages?

7. The attention span of a two-year-old is exponentially distributed with a mean of about eight minutes. Suppose we randomly survey 60 two-year-olds.

a. In words, ? = _______

b. ? ~ _____(_____,_____)

c. In words, X ¯ = ____________

d. X ¯ ~ _____(_____,_____)

e. Calculate the probabilities:

i. The probability that an individual attention span is less than ten minutes.

ii. The probability that the average attention span for the 60 children is less than ten minutes?

Chapter 8:

8. Among various ethnic groups, the standard deviation of heights is known to be approximately three inches. We wish to construct a 95% confidence interval for the mean height of male Swedes. Forty-eight male Swedes are surveyed. The sample mean is 71 inches. The sample standard deviation is 2.8 inches.

a. i. x ¯ =________ ii. ? =________ iii. n =________

b. In words, define the random variables X and X ¯ .

c. Which distribution should you use for this problem? Explain your choice.

d. Construct a 95% confidence interval for the population mean height of male Swedes.

i. State the confidence interval. ii. Sketch the graph. iii. Calculate the error bound.

e. What will happen to the level of confidence obtained if 1,000 male Swedes are surveyed instead of 48? Why?

9. Suppose that an accounting firm does a study to determine the time needed to complete one person’s tax forms. It randomly surveys 100 people. The sample mean is 23.6 hours. There is a known standard deviation of 7.0 hours. The population distribution is assumed to be normal.

a. i. x ¯ =________ ii. ? =________ iii. n =________

b. In words, define the random variables X and X ¯ .

c. Which distribution should you use for this problem? Explain your choice.

d. Construct a 90% confidence interval for the population mean time to complete the tax forms.

i. State the confidence interval. ii. Sketch the graph. iii. Calculate the error bound.

e. If the firm wished to increase its level of confidence and keep the error bound the same by taking another survey, what changes should it make?

f. If the firm did another survey, kept the error bound the same, and only surveyed 49 people, what would happen to the level of confidence? Why?

g. Suppose that the firm decided that it needed to be at least 96% confident of the population mean length of time to within one hour. How would the number of people the firm surveys change? Why?

10. A camp director is interested in the mean number of letters each child sends during his or her camp session. The population standard deviation is known to be 2.5. A survey of 20 campers is taken. The mean from the sample is 7.9 with a sample standard deviation of 2.8.

a. i. x ¯ =________ ii. ? =________ iii. n =________

b. Define the random variables X and X ¯ in words.

c. Which distribution should you use for this problem? Explain your choice.

d. Construct a 90% confidence interval for the population mean number of letters campers send home.

i. State the confidence interval. ii. Sketch the graph. iii. Calculate the error bound.

e. What will happen to the error bound and confidence interval if 500 campers are surveyed? Why?

Answer 1