Exam 10: Hypothesis Tests Involving a Sample Mean or Proportion

Soft drink A soft drink dispenser is set to automatically dispense drinks and ice in 8 oz.cups.The machine is set to fill the cups with 7 1/4 oz.so that the liquid will not run over the top of the cup.The company is getting complaints that the machine is filling the cups too full and spilling over.When the machine is operating properly,only 1% of the cups will not be filled properly.The production manager conducted a one-tail hypothesis test at the 0.01 level to check the machine.A sample of 50 fills was measured. -A hypothesis test has been set up and is to be conducted at the $\alpha$ =.025 level of significance.If the sample is doubled,what will be the effect on $\alpha$ ? On $\beta$ ?

Based on sample data,the 90% confidence interval limits for the population mean are 170.86 and 195.42.If the 10% level of significance was used in testing the hypothesesH₀: $\mu$ = 201,H₁: $\mu \neq$ 201,the null hypothesis:

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. -n = 225; p = 0.13; H₀: $\pi$ = 0.10,H₁: $\pi \neq$ 0.10; $\alpha$ = 0.02 State the conclusion:

Adult drink A particular study claims that the average U.S.adult drinks 1.7 cups of coffee per day.To test this claim,a sample of 35 adults were asked about their coffee consumption.The sample average was 1.95 cups per day.Assume the population standard deviation is 0.5 cups per day and set $\alpha$ = 0.10. -State the decision rule in terms of the critical value.

If a hypothesis is rejected at the .025 level of significance,it:

Directional assertions lead to what is called a two-tail test.

Local Restaurant A local restaurant claimed in an advertisement that at least 80% of their customers rate their establishment as "Excellent".An independent survey company asked 200 customers their opinion and from this random sample,168 customers rated the establishment "Excellent".Use $\alpha$ = 0.05. -State the null and alternative hypotheses.

Sermons My daughter and I have argued about the average length of our preacher's sermons on Sunday morning.Despite my arguments,she thinks that the sermons are more than twenty minutes (over twenty minutes is important to her especially when soccer was playing on TV).For one year she has randomly selected 12 Sundays and found the average time of 26.42 minutes with a standard deviation of 6.69 minutes.Assuming that the population is normally distributed and using a 0.05 level of significance,is my daughter correct in thinking that the average length of sermons is more than 20 minutes? -State the null and alternative hypotheses. H₀: ____________________ H₁: ____________________

If a test is carried out at a specified significance level (e.g., $\alpha$ = .05),using a larger sample size will reduce the value of $\beta$ ,the probability of making a Type II error.

Suppose that we reject a null hypothesis at the .05 level of significance.Then for which of the following $\alpha$ values do we also reject the null hypothesis?

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 conclusion?

Sermons My daughter and I have argued about the average length of our preacher's sermons on Sunday morning.Despite my arguments,she thinks that the sermons are more than twenty minutes (over twenty minutes is important to her especially when soccer was playing on TV).For one year she has randomly selected 12 Sundays and found the average time of 26.42 minutes with a standard deviation of 6.69 minutes.Assuming that the population is normally distributed and using a 0.05 level of significance,is my daughter correct in thinking that the average length of sermons is more than 20 minutes? -What is the decision rule?

Retail store A retail store claims that the average age of their customers is less than 40 years old.A sample of 50 customers had an average age of 38.7 years.Assume the population standard deviation is 12.5 years and use $\alpha$ = 0.05. -State the null and alternative hypothesis.

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 decision rule?

A social scientist claims that the average adult watches less than 26 hours of television per week.He collects data on 25 individuals' television viewing habits and finds that the mean number of hours that the 25 people spent watching television was 22.4 hours.If the population standard deviation is known to be eight hours,can we conclude at the 1% significance level that he is right? Appropriate test: ____________________ Test statistic = ____________________ Conclusion: ______________________________

General Manager The general manager of a large discount chain was considering location of a new store in a small mid-western city.The decision was dependent upon the per capita retail sales for the city being significantly greater than $4500.A sample of 500 residents was taken,and showed a mean of $4780.A population standard deviation of $2500 was known. -Determine the p-value associated with each of the following values of the standardized test statistic z. A)Two-tail test,with z = 1.50 B)One-tail test,with z = 1.05 C)One-tail test,with z = -2.40

The power of a test is the probability of making:

Tail graph -Using the above graph,choose the answers that indicate a two-tail test: Area 1: ____________________ Area 2: ____________________ Area 3: ____________________

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 decision rule is: Reject H₀ if the calculated t > 2.68 or if t < -2.68.

Soft drink A soft drink dispenser is set to automatically dispense drinks and ice in 8 oz.cups.The machine is set to fill the cups with 7 1/4 oz.so that the liquid will not run over the top of the cup.The company is getting complaints that the machine is filling the cups too full and spilling over.When the machine is operating properly,only 1% of the cups will not be filled properly.The production manager conducted a one-tail hypothesis test at the 0.01 level to check the machine.A sample of 50 fills was measured. -For the critical z,identify the sample proportion that this z-value represents. p = ____________________ Use the value of p determined above to state a decision rule for the test.