Question 1.

A data set of size 500 was obtained. Each value was one of the numbers 1,2,3,4, or 5, and the data are summarized as follows:

Determine the sample mean:

Question 2.

From the table above, calculate the median:

Question 3.

From the table above, calculate the mode.

Question 4.

From the table above, calculate the sample variance, \({{S}^{2}}\).

Question 5.

The following data resulted from independent measurements of the melting point of lead (in degrees Centigrade):

Assuming that the measurements can be regarded as constituting a normal sample whose mean is the true melting point of lead, determine a 95% two-sided confidence interval for this value.

Enter the lower bound on the 95% CI:

Question 6.

From the data above, enter the upper bound on the 95% CI:

Question 7.

A random sample of 49 5-year-olds was surveyed, and the average number of tantrums they had thrown in the past week was 1.9. The standard deviation is known to be 0.5.

1. Construct a 95% confidence interval for the average number of tantrums thrown by 5-year-olds in a week.
   Enter the lower bound on the 95% CI:
   Question 8.
   From the data above, enter the upper bound on the 95% CI:
   Q uestion 9.
2. Your boss respectfully requests that you rework this problem so that your confidence interval is absolutely no more than ± 0.1. (The total width of the interval should be no more than 0.2.) How large a sample should you now poll in order to please your boss?

Question 10.

An article in the Journal of Composite Materials (Dec. 1989, vol. 23, p. 1200) describes the effect of delamination on the natural frequency of beams made from composite laminates. Five such delaminated beams were subjected to loads, and the resulting frequencies were as follows (in Hz):

230.66, 233.05, 232.58, 229.48, 232.58

Does this data support the claim that the mean natural frequency is 235 Hz? Assume \(\alpha \) = 0.05 and enter the critical value.

Question 11.

A machine produces metal rods used in an automobile suspension system. A random sample of 15 rods is selected, and the diameter is measured. The resulting data are shown below:

Is there sufficient evidence to indicate that the mean rod diameter exceeds 8.20 mm using \(\alpha =\) 0.05?

1. What is the appropriate critical value?
   Question 12.
2. What is the t-value of this sample?
   Question 13.
3. Should you reject the null hypothesis? (yes or no)

Question 14.

An anthropologist wishes to estimate the average height of men for a certain race of people. If the population standard deviation is assumed to be 2.5 inches, and if she randomly samples 100 men, what is the probability that the difference between the sample mean and the true population mean will not exceed 0.5 inches?

Question 15.

A company's manufacturing process has reliably been shown to produce ceiling fans that have X defects per ceiling fan, where X ~ Normal(μ = 10, _σ 2= 16). A new, up and coming director implements an "improved" process. A sample of size n =25 ceiling fans is collected, and we observe that the average number of defects per fan is 8.60. On the basis of this sample, answer the following questions (in order):

1. What is the value of the test statistic for the sample of the sample?
   Question 16.
2. At the 5% level of significance, what is the appropriate critical level?
   Question 17.
3. At the 1% level of significance, what is the appropriate critical level?
   Question 18.
4. Would you conclude, at the 5% level of significance, that the process now produces fewer defects per fan? (yes or no)
5. Would you conclude, at the 1% level of significance, that the process now produces fewer defects per fan? (yes or no)

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