Exam 10: Hypothesis Tests Involving a Sample Mean or Proportion

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. -State the decision rule in terms of the critical value.

Identify the following equation: $z = \left( \mathrm { p } - \pi _ { 0 } \right) / \sigma _ { p }$

What is an alternative hypothesis?

Tobacco Industry The tobacco industry advertised that the per capita number of cigarettes sold per year was 3700.An anti-smoking organization argued strongly that per capita consumption had declined. -State the decision rule for $\alpha = 0.05$

Failure to reject a null hypothesis constitutes proof that it is true.

A portfolio manager was analyzing the price-earnings ratio for this year's performance.His boss said that the average price-earnings ratio was 20 for the many stocks,which his firm had traded,but he felt that figure was too high.The portfolio manager randomly selected a sample of 50 P/E ratios and found a mean of 18.17 and standard deviation of 4.60.Assume that the population is normally distributed and test at the 0.01 level of significance. The correct hypotheses are: H₀: $\mu \geq$ 20,H₁: $\mu$ < 20.

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. -Determine the lower and upper critical values for $\bar { x }$ . Lower value = ____________________ Upper value = ____________________

Reducing the probability of a Type II error also reduces the probability of a Type I error.

A researcher wants to carry out a hypothesis test involving the mean for a sample of size n = 15.He does not know the true value of the population standard deviation,but is reasonably sure that the underlying population is approximately normally distributed.Should he use a z-test or a t-test in carrying out the analysis? Why?

Use the given information to calculate the p-value and interpret the result: H₀: $\mu$ = 60,H₁: $\mu$ < 60 $\text { σ }$ = 5,n = 100, $\bar { x }$ = 59, $\alpha$ = 0.05 p-value = ____________________ Interpretation: ____________________

The conclusion from a non-directional hypothesis test for a population mean at the $\alpha$ level of significance will be the same as the conclusion based on a confidence interval at the 100 (1 - $\alpha$ )% confidence level.

What is an operating characteristic curve and how is it related to the power curve for a test?

The rejection and non-rejection regions can be stated as a decision rule specifying the conclusion to be reached for a given outcome of the test.

Confidence intervals and hypothesis tests are both concerned with using sample information to make a statement about the unknown value of a population parameter.Thus,it is not surprising that their results are related.

It has been claimed that no more than 3% of the units coming off an assembly line are defective.Will this test be a one-tail - left,a one tail - right,or a two-tail? Explain your answer.

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? -What is your conclusion? Appropriate test: ____________________ Test statistic = ____________________ Conclusion: ____________________

County Agricultural A county agricultural analyst took a sample of average annual precipitation for 50 cities.He wanted to test the null hypothesis that the average annual rainfall was at most 31.0 inches against the alternative that it was greater than 31.0 inches.Assume that the sample standard deviation was 13.76,and the sample mean was 34.74.Perform the appropriate test for the county analyst at the 0.05 level. -What is the interpretation of your conclusion?

Light Bulb Mnf. A light bulb manufacturer claims that the average life of their light bulb is more than 1000 hours.To test this claim,a sample of 32 light bulbs had an average life of 1120 hours.Assume the population standard deviation is 325 hours and use $\alpha$ = 0.02. -State the null and alternative hypothesis.

A mortgage broker is offering home mortgages at a rate of 9.5% but is fearful that this value is higher than many others are charging.A sample of 40 mortgages filed in the county court house shows an average of 9.25% with a standard deviation of 8.61%.Does this sample indicate a smaller average? Use $\alpha$ = 0.05 and assume a normally distributed population.

A Type I error is represented by the Greek letter ____________________,which represents the probability of rejecting a ____________________.