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

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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.

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