Foundations of Business Statistics Professor Troudt

Foundations of Business Statistics – Professor Troudt Materials from Professor Friedman

Christina Upshaw HOMEWORK: One-Sample Z-test and Confidence Intervals (Two-sided)

PROBLEM 1:

A company claims that its soup machines deliver exactly 10.0 ounces of soup—no more, no less. A researcher samples 100 bowls of soup and finds that: X = 10.40 ounces s = 1.20 ounces (a) Test the company’s claim at a .05 significance level

H 0 :m= 10 H1:m 10

x -m 10.4 - 10 Test statistic Z = = = 3.333 s n 1.2 100

Two-sided critical value |z| > 1.96

Since 3.33 > 1.96 we reject H0 and conclude that the soup machines do not deliver exactly 10.0 ounces of soup.

Topic 6B: One Sample Tests p. 1 Foundations of Business Statistics – Professor Troudt Materials from Professor Friedman

PROBLEM 2: A company claims that its toasters have an average life of at least 10 years. Use the following sample to test at  = .02. n=100 X = 9.4 years s = 1.2 years

H 0 :m 10 H1:m< 10

x -m 9.4 - 10 Test statistic Z = = = - 5 s n 1.2 100

One-sided critical value z < -2.054

Since -5 < 2.054 we reject H0 and conclude that the toasters do not have an average life of at least 10 years.

PROBLEM 3: A manufacturer produces drill bits with an intended life of at least 580 hours and a standard deviation of 30 hours. A quality control scientist draws a sample of 100 bits and finds X =577. Test at α=.05 to see if the machinery needs adjusting.

H 0 :m 580 H1:m< 580

x -m 577 - 580 Test statistic Z = = = - 1 s n 30 100 critical value z < -1.645 (one-sided, α=.05)

Since -1 > -1.645 we fail to reject H0 and conclude that the machinery does not deviate too much from the specifications so it does not need adjusting.

PROBLEM 4:

A researcher samples 100 Dell laptops and finds: X = 6.10 years s = 0.90 years (a) Construct a 95% confidence interval

95% CI:

s m= x� z0.05/ 2 n 0.9 6.1� 1.96 100 6.1� 0.1764 (5.9236, 6.2764)

PROBLEM 5:

A company that manufactures batteries is interested in constructing a two-sided 95% confidence interval for the population mean. They sampled 121 batteries and found that the sample mean is 97.0 hours and the sample standard deviation is 3 hours.

95% CI: s m= x� z0.05/2 n 3 97� 1.96 121 97� 0.5345 (96.4655, 97.5345)

PROBLEM 6:

A company that manufactures toasters is interested in constructing a two-sided 98% confidence interval for the population mean. They sampled 100 toasters and found that the sample mean is 9.40 years and the sample standard deviation is 1.20 years.

98% CI:

s m= x� z0.02/ 2 n 1.2 9.4� 2.326 100 9.4� 0.27912 (9.12088, 9.67912)

PROBLEM 7:

A college wants to know the upper limit of a two-sided 95% confidence interval for SAT scores on the Math section (range equals 200 to 800). They sampled 400 students and found that the sample mean was 550 and the sample standard deviation was 80.

95% CI s m= x� z0.05/2 n 80 550� 1.96 400 550� 7.84 (542.16, 557.84)

PROBLEM 8:

The state of Alabama wants to know the lower limit of a two-sided 90% confidence interval for reading scores of graduating seniors at all high schools in Alabama. They sampled 900 students and found that the sample mean was 11.30 and the sample standard deviation was 1.50

s 1.5 Lower Limit: m>x -z0.10/2 = 11.30 - 1.645 = 11.21775 n 900

PROBLEM 9:

A company claims that people using its sleeping pill will sleep for at least 8 hours a night. (a) Use the following sample to test at  = .05. (b) suppose no claim was made but the company is thinking of making a claim using a 95% two-sided confidence interval. Construct the interval using the same data and suggest the maximum claim the company can make. n=144 X = 7.5 hours s = 1.8 hours

H 0 :m 8 H1:m< 8

x -m 7.5 - 8 Test statistic Z = = =- 3.333 s n 1.8 144

One-sided critical value z < -1.645

Since -3.33 < -1.645 we reject H0 and conclude that the average sleeping time is not greater than or equal to 8 hrs/night.

95% CI: s m= x� z0.05/2 n 1.8 7.5� 1.96 144 7.5� 0.294 ( 7.206, 7.794)

They maximum claim is 7.794 hrs of sleep at night.

Topic 6B: One Sample Tests p. 9