Question

3. Estimate Interval The makers of a soft drink want to identify the average age of its consumers. A sample of 61 consumers was taken. The average age in the sample was 23 years with a sample standard deviation of 5 years. Please answer the following questions: a. Construct a 95% confidence interval estimate for the mean of the consumers’ age. b. Suppose a sample of 85 was selected (with the same mean and the sample standard deviation). Construct a 95% confidence interval for the mean of the consumers’ age.

[Hint: Please see Chap008 – Slides 24-29 for formula and example. Please also see page 343-349 in the textbook.]

4. Hypothesis Testing Annual per captial consumption of milk is 21.6 gallons (Statistical Abstract of the United States: 2006). Being from the Midwest, you believe milk consumption is higher there and wish to support your opinion. A sample of 16 individuals from the Midwestern town of Webster City showed a sample mean annual consumption of 24.1 gallons with a sample standard deviation of s=4.8. a. Develop a hypothesis test that can be used to determine whether the mean annual consumption in Webster City is higher than the national mean. b. At α=0.05, test for a significant difference. What is your conclusion? Extra credit 5. A lathe is set to cut bars of steel into lengths of 9 centimeters. The lathe is considered to be in perfect adjustment if the average length of the bars it cuts is 9 centimeters. A sample of 100 bars is selected randomly and measured. It is determined that the average length of the bars in the sample is 9.085 centimeters. Suppose the population standard deviation is 0.335 centimeters. a. Formulate the hypotheses to determine whether or not the lathe is in perfect adjustment. b. Compute the test statistic. c. Using the p-value approach, what is your conclusion? Let α = .05.

Answer 1

3) The makers of a soft drink want to identify the average age of its consumers. A sample of 61 consumers was taken. The average age in the sample was 23 years with a sample standard deviation of 5 years. Please answer the following questions: a. Construct a 95% confidence interval estimate for the mean of the consumers’ age. b. Suppose a sample of 85 was selected (with the same mean and the sample standard deviation). Construct a 95% confidence interval for the mean of the consumers’ age.

a) Given:

(Here z = 1.96 because it is the score for 95% Confidence Interval)

95% confidence interval estimate for the mean of the consumers’ age is (17.22,28.77)

b) Given:

  

  

95% confidence interval for the mean of the consumers’ age is (18.11,27.88)

4) Hypothesis Testing Annual per captial consumption of milk is 21.6 gallons (Statistical Abstract of the United States: 2006). Being from the Midwest, you believe milk consumption is higher there and wish to support your opinion. A sample of 16 individuals from the Midwestern town of Webster City showed a sample mean annual consumption of 24.1 gallons with a sample standard deviation of s=4.8. a. Develop a hypothesis test that can be used to determine whether the mean annual consumption in Webster City is higher than the national mean. b. At α=0.05, test for a significant difference. What is your conclusion?

a) Develop a hypothesis test that can be used to determine whether the mean annual consumption in Webster City is higher than the national mean.

b) At α=0.05, test for a significant difference. What is your conclusion?

Given:  

x-bar = 24.1 s = 4.8 n = 16

The test statistic is

Given alpha = 0.05,

The critical value is [t(0.05, df=n-1=15)]=1.75 (check student t table)

Since t=2.08 is larger than 1.75, we reject Ho.

Hence we conclude that  the mean annual consumption in Webster City is higher than the national mean.

5) A lathe is set to cut bars of steel into lengths of 9 centimeters. The lathe is considered to be in perfect adjustment if the average length of the bars it cuts is 9 centimeters. A sample of 100 bars is selected randomly and measured. It is determined that the average length of the bars in the sample is 9.085 centimeters. Suppose the population standard deviation is 0.335 centimeters. a. Formulate the hypotheses to determine whether or not the lathe is in perfect adjustment. b. Compute the test statistic. c. Using the p-value approach, what is your conclusion? Let α = .05.

a. Formulate the hypotheses to determine whether or not the lathe is in perfect adjustment.

b. Compute the test statistic.

Given:

n = 100 x-bar = 9.085 s = 0.335 mu=9

c. Using the p-value approach, what is your conclusion?

P-value is 0.0127.

0.0127 < 0.05

Hence we reject null hypothesis.

Conclusion:

It appears that the mean length is different from 9 cm.