Exam 10: Hypothesis Tests Involving a Sample Mean or Proportion

Dimensions The dimensions of a two by four board are not actually two inches by four inches.Rather the actual dimensions are supposed to be 1.50 inches by 3.50 inches.A planer finishes the boards to the proper dimensions.To maintain quality control,periodic checks are made to see that the planer is set to the correct dimension.When the planer is adjusted properly,the average width is $\mu$ = 3.5 inches.If the average width is not 3.5 inches,the planer must be adjusted.Thus the null hypothesis is $\mu$ = 3.5 inches and the alternate hypothesis is $\mu \neq$ 3.5 inches.For control purposes,a random sample of 49 boards is taken twice daily.The standard deviation of the process is $\text { σ }$ = 0.25 inches.To minimize the unnecessary adjustment of the planer,the risk of a Type I error, $\alpha$ ,is set at 0.05. -State the decision rule.

The p-value of a test is the level of significance,where the observed value of the test statistic is exactly the same as a critical value for that level.

A non-directional assertion involves what are called one-tail tests.

Golf A round of golf on your typical course requires at least 4.5 hours.A particular golf club claims that the average length of time to complete a round of golf on their course is less than 4.5 hours.To test this claim,the club sampled 15 golfers and recorded their time to finish their round.The sample average was 4.4 hours with a standard deviation of 0.8 hours.Assume the time to complete a round of golf is normally distributed and use $\alpha$ = 0.01. -A random sample of 7 observations was drawn from a normal population.The following summations were computed: $\Sigma$ x_i = 63.5 and $\Sigma$ x_i ² = 580.2. Test the following hypotheses: H₀: $\mu$ = 8,H₁: $\mu$ > 8,at the 1% significance level. Appropriate test:____________________ Test statistic = ____________________ Conclusion: ____________________

If we reject the null hypothesis,we conclude that:

Use the given information to calculate the p-value and interpret the result: H₀: $\mu$ = 40,H₁: $\mu \neq$ 40 $\text { σ }$ = 5.5,n = 25, $\bar { x }$ = 42, $\alpha$ = 0.10 p-value = ____________________ Interpretation: ____________________

Table 1 Based on the sample data,a confidence interval has been constructed such that we have 95% confidence that the population mean is between 75 and 105. -What conclusion would be reached at the $\alpha$ = .05 level in testing H₀: $\mu$ = 125 versus H₁: $\mu \neq$ 125?

Table 1 Based on the sample data,a confidence interval has been constructed such that we have 95% confidence that the population mean is between 75 and 105. -What conclusion would be reached at the $\alpha$ = .05 level in testing H₀: $\mu$ = 100 versus H₁: $\mu \neq$ 100?

In order to determine the p-value,which of the following is not needed?

During the last energy crisis,a government official claimed that the average car owner refilled the tank when there was more than 3 gallons left.To check the claim,10 cars were surveyed as they entered a gas station.The amount of gas was measured and recorded as follows: 3,5,3,2,3,3,2,6,4,and 1.If we assume that the amount of gas remaining in tanks is normally distributed with a standard deviation of one gallon,can we conclude at the 10% significance level that the official was correct? Appropriate test: ____________________ Test statistic = ____________________ Conclusion: _________________________ Interpretation: _____________________________________________

A long-haul truck driver is eligible for a year-end bonus if the average miles driven per day are more than 375 miles per eight-hour day.To determine whether or not a driver is to receive a year-end bonus,a random sample of 36 days from the driver's logbook is taken and the sample mean $\bar { x }$ is computed.The hypotheses are: H₀: $\mu \leq$ 375 miles (no bonus)vs.H₁: $\mu$ > 375 miles (eligible for bonus).If the population standard deviation $\text { σ }$ = 70 miles and the true mean is 405,what is the power of the test at $\alpha$ = 0.05?

Bank A bank holds the mortgage on a motel which the motel owner says has an average daily revenue of at least $2048.In the event the average is not at least $2048,the bank will ask the motel owner for additional collateral for the loan as protection against loan default.A loan officer is aware that the daily average revenue is normally distributed and randomly selects 20 days during the last six months.The sample has a mean of $2003 and standard deviation of $144.00.At the 0.01 level,is the bank loan financially sound (should the bank ask for additional collateral)? What is the value of test statistic t?

Life span Research has claimed that the average life span of a male mosquito is 15 days.To test this claim,the life spans of 20 male mosquitos were observed to have an average of 16.5 days and a standard deviation of 2.2 days.Assume that the life spans are normally distributed and set $\alpha$ = 0.05. -State the decision rule in terms of the critical value.

Restaurant A restaurant claims that the average gratuity per table is $15.00.To test this claim,a sample of 18 tables was selected which had an average gratuity of $13.75 with a standard deviation of $2.60.Assume the population of gratuities is normally distributed and use $\alpha$ = 0.10. -State the decision rule in terms of the critical value.

For a two-tail hypothesis test,the level of significance is the sum of both tail areas.

Life span Research has claimed that the average life span of a male mosquito is 15 days.To test this claim,the life spans of 20 male mosquitos were observed to have an average of 16.5 days and a standard deviation of 2.2 days.Assume that the life spans are normally distributed and set $\alpha$ = 0.05. -State the null and alternative hypotheses.

Consider testing the hypothesis H₀: $\mu$ = 800 vs.H₁: $\mu \neq$ 800. If the value of the test statistic z equals 1.75,then the p-value is:

Political candidate A political candidate needs more then 50% of the votes to win the next election.In an effort to convince contributors to his campaign,he samples 800 potential voters of which 432 indicated they would vote for him.Use $\alpha$ = 0.05. -State the decision rule in terms of the critical value.

A directional claim or assertion states that a population parameter is greater than (>),at least,( $\ge$ ),no more than( $\le$ ),or less than (<),some quantity.

Merit In order to receive a merit raise a computer clerk must average less than 15 minutes per order form processed.An across-the-board raise was given if the average is 15 minutes or more.The supervisor randomly selects 36 order forms and carefully times the length of time required to enter all of the data on each form.The significance level = 0.02.The average time to enter the 36 order forms is 14.36 minutes,and the population standard deviation is 3.03 minutes with a normal distribution.The test question is: Should a merit raise be given? -State the decision rule in terms of the critical value as well as the p-value.