# Population mean, one-tail test, p-value, standard deviation

This assignment has 9 questions to be answered. Some questions having multiple sub-questions. Please be sure to review the attachment.

10.20 If the population standard deviation is known,

but the sample size is less than 30, what assumption is

necessary to use the z-statistic in carrying out a hypothesis

test for the population mean?

10.18 It has been claimed that no more than 5% of

the units coming off an assembly line are defective.

Formulate a null hypothesis and an alternative

hypothesis for this situation. Will the test be one-tail

or two-tail? Why? If the test is one-tail, will it be

left-tail or right-tail? Why?

10.28 For a sample of 12 items from a normally

distributed population for which the standard deviation

is 17.0, the sample mean is 230.8. At the 0.05 level

of significance, test H0;greater than or equal to 220 versus H1 .

Determine and interpret the p-value for the test.

10.29 A quality-assurance inspector periodically

10.44 The International Coffee Association has reported

the mean daily coffee consumption for U.S. residents as

1.65 cups. Assume that a sample of 38 people from a

North Carolina city consumed a mean of 1.84 cups of

coffee per day, with a standard deviation of 0.85 cups. In a

two-tail test at the 0.05 level, could the residents of this

city be said to be significantly different from their counterparts

across the nation?

10.74 It has been reported that 80% of taxpayers who

are audited by the Internal Revenue Service end up

paying more money in taxes. Assume that auditors are

randomly assigned to cases, and that one of the ways

the IRS oversees its auditors is to monitor the

percentage of cases that result in the taxpayer paying

more taxes. If a sample of 400 cases handled by an

individual auditor has 77.0% of those she audited paying

more taxes, is there reason to believe her overall

"pay more" percentage might be some value other than

80%? Use the 0.10 level of significance in reaching a

conclusion. Determine and interpret the p-value for the

test.

11.70 According to the National Association of Homebuilders,

the average life expectancies of a dishwasher

and a garbage disposal are about the same: 10 years.

Assume that their finding was based on a sample of

n1 = 60 dishwashers and n2 = 40 garbage disposals, and

that the corresponding sample standard deviations were

s1 = 3.0 years and s2 = 3.7 years. Using the 0.02 level of

significance, examine whether the population standard

deviations for the lifetimes of these two types of

appliances could be the same.

Please be sure to review and answer the questions on the attachment.

#### Solution Preview

See the attached file.

10.20 If the population standard deviation is known, but the sample size is less than 30, what assumption is necessary to use the z-statistic in carrying out a hypothesis test for the population mean?

The assumption needed is that the population distribution is normal or approximately normal.

10.18 It has been claimed that no more than 5% of the units coming off an assembly line are defective. Formulate a null hypothesis and an alternative hypothesis for this situation. Will the test be one-tail or two-tail? Why? If the test is one-tail, will it be left-tail or right-tail? Why?

H0: The proportion of units coming out of the assembly >= 0.05

H1: The proportion of units coming out of the assembly < 0.05

Please note that whatever we claim should be written as an alternate hypothesis.

This is a one tail test, as we want to establish that the production system is in good shape and does not produce more than 5% defective items. However, if it produces significantly less than 5% we are not bothered.

10.28 For a sample of 12 items from a normally

distributed population for which the standard deviation

is 17.0, the sample mean is 230.8. At the 0.05 level

of significance, test H0;greater than or equal to 220 versus H1 .

Determine and interpret the p-value for the test.

Sample Mean 230.8

Population Standard Deviation 17

Sample size 12

Standard Error 4.91

This is a one tail test as we are testing whether the sample mean is less than 220 or not. It is a z-test as although the sample size is less than 30, we know the standard deviation and the population distribution is normal.

Level of significance 0.05

Z-observed 2.200723379

Z-critical 1.645

Since z-critical is less than z-observed, we reject the null hypothesis and conclude that mean is statistically significantly greater than 220.

p-value 0.0139

The p-value of 0.0139 indicates that if the significance level is 0.0139 or above, we will reject the null hypothesis and conclude that the mean is statistically higher than 220.

10.44 The International Coffee ...

#### Solution Summary

Population mean, one-tail test, p-value and standard deviation is examined.