Exam 9: Hypothesis Testing

For a given sample size,any attempt to reduce the likelihood of making one type of error (Type I or Type II)will increase the likelihood of the other error.

A car dealer who sells only late-model luxury cars recently hired a new salesman and believes that this salesman is selling at lower markups.He knows that the long-run average markup in his lot is $5600.He takes a random sample of 16 of the new salesman's sales and finds an average markup of $5000 and a standard deviation of $800.Assume the markups are normally distributed.What is the value of an appropriate test statistic for the car dealer to use to test his claim?

A fast-food franchise is considering building a restaurant at a busy intersection.A financial advisor determines that the site is acceptable only if,on average,more than 300 automobiles pass the location per hour.If the advisor tests the hypotheses versus ,µ stands for ____________.

A university wants to know if the average salary of its graduates has increased since 2010.The average salary of graduates prior to 2010 was $48,000.Since 2010,the university surveyed 256 graduates and found an average salary of $48,750.Assume that the standard deviation of all graduates' salaries is $7,000. A)Specify the null and alternative hypotheses to determine whether the average salary of graduates has increased. B)Calculate the value of the test statistic and the critical value at a 5% significance level. C)At the 5% significance level,can you conclude that salaries have increased? Explain.

Exhibit 9-2.The owner of a large car dealership believes that the financial crisis decreased the number of customers visiting her dealership.The dealership has historically had 800 customers per day.The owner takes a sample of 100 days and finds the average number of customers visiting the dealership per day was 750.Assume that the population standard deviation is 350. Refer to Exhibit 9-2.At a 5% significance level,the critical value(s)is (are)___________.

Under the assumption that the null hypothesis is true as an equality,the p-value is the likelihood of observing a sample mean that is at least as extreme as the one derived from the given sample.

You want to test if more than 20% of homes in a neighborhood have recently been sold through a short sale,at a foreclosure auction,or by the bank following an unsuccessful foreclosure auction.You take a sample of 60 homes from this neighborhood and find that 14 fit your criteria.The appropriate null and alternative hypotheses are ___________.

Billy wants to test whether the average speed of his favorite pitcher's fastball differs from the league average of 92 miles per hour.He takes a sample of 36 of the pitcher's fastballs and computes a sample mean of 94 miles per hour.Assume that the standard deviation of the population is 4 miles per hour. A)Specify the null and alternative hypotheses to test Billy's claim. B)Calculate the value of the test statistic and the p-value. C)At the 5% significance level,can you conclude that Billy's favorite pitcher's fastball differs in speed from the league average? D)At the 1% significance level,can you conclude that Billy's favorite pitcher's fastball differs in speed from the league average?

The null hypothesis typically corresponds to a presumed default state of nature.

Exhibit 9-3.The Boston public school district has had difficulty maintaining on-time bus service for its students ("A Year Later,School Buses Still Late," Boston Globe,October 5,2011).Suppose the district develops a new bus schedule to help combat chronic lateness on a particularly woeful route.Historically,the bus service on the route has been,on average,12 minutes late.After the schedule adjustment,the first 36 runs were an average of 8 minutes late.As a result,the Boston public school district claimed that the schedule adjustment was an improvement-students were not as late.Assume a population standard deviation for bus arrival time of 12 minutes. Refer to Exhibit 9-3.At the 5% significance level,does the evidence support the Boston public school district's claim?

An engineer wants to know if the average amount of energy used in his factory per day has changed since 2005.The factory used an average of 2000 megawatt hours (mwh)per day prior to 2005.Since 2005,the engineer surveyed 400 days and found the average energy use was 2040 mwh per day.Assume that the population standard deviation is 425 mwh per day. A)Specify the null and alternative hypotheses to determine whether the factory's energy use has changed. B)Calculate the value of the test statistic and the p-value. C)At the 5% significance level,can you conclude that the energy usage has changed? Explain.

A hairdresser believes that she is more profitable on Tuesdays,her lucky day of the week.She knows that,on average,she has a daily revenue of $250.She randomly samples the revenue from eight Tuesdays and finds she takes in $260,$245,$270,$260,$295,$235,$270,and $265.Assume that daily revenue is normally distributed. A)Specify the population parameter to be tested. B)Specify the null and alternative hypotheses to test the hairdresser's claim. C)Calculate the sample mean revenue and the sample standard deviation. D)Compute the value of the appropriate test statistic. E)At the 10% significance level,specify the critical value(s). F)At the 10% significance level,is the hairdresser's claim supported by the data?

Students who graduated from college in 2010 with student loans owed an average of $25,250 (The New York Times,November 2,2011).An economist wants to determine if the average debt has increased since 2010.She takes a sample of 40 recent graduates and finds that their average debt was $28,275.Assume that the population standard deviation is $7,250. A)Specify the competing hypotheses to determine whether the average undergraduate debt has increased since 2001. B)Calculate the value of the test statistic and the p-value. C)At the 5% significance level,can you conclude that the average undergraduate debt has increased? Explain.

Exhibit 9-7.Vermont-based Green Mountain Coffee Roasters dominates the market for single-serve coffee in the United States,with its subsidiary Keurig accounting for approximately 70% of sales ("Rivals Try to Loosen Keurig's Grip on Single-Serve Coffee Market," Chicago Tribune,February 26,2011).But Keurig's patent on K-cups,the plastic pods used to brew the coffee,is expected to expire in 2012,allowing other companies to better compete.Suppose a potential competitor has been conducting blind taste tests on its blend and finds that 47% of consumers strongly prefer its French Roast to that of Green Mountain Coffee Roasters.After tweaking its recipe,the competitor conducts a test with 144 tasters,of which 72 prefer its blend.The competitor claims that its new blend is preferred by more than 47% of consumers to Green Mountain Coffee Roasters' French Roast. Refer to Exhibit 9-7.What critical value should be used to test this claim at the 1% significance level?

When we reject the null hypothesis when it is actually false we have committed _________.

Exhibit 9-3.The Boston public school district has had difficulty maintaining on-time bus service for its students ("A Year Later,School Buses Still Late," Boston Globe,October 5,2011).Suppose the district develops a new bus schedule to help combat chronic lateness on a particularly woeful route.Historically,the bus service on the route has been,on average,12 minutes late.After the schedule adjustment,the first 36 runs were an average of 8 minutes late.As a result,the Boston public school district claimed that the schedule adjustment was an improvement-students were not as late.Assume a population standard deviation for bus arrival time of 12 minutes. Refer to Exhibit 9-3.At the 1% significance level,does the evidence support the Boston Public Schools' claim?

In a one-tailed test,the rejection region is located under one tail (left or right)of the corresponding probability distribution,while in a two-tailed test this region is located under both tails.

According to a 2009 Lawyers.com survey,only 35% of adult Americans had a will (The Wall Street Journal,December 12,2011).Suppose a recent survey of 250 adult Americans found 100 adults with wills. A)Specify the competing hypotheses to determine whether the proportion of adult Americans that have a will differs from the proportion reported for 2009. B)Calculate the value of the test statistic and the p-value. C)At the 5% significance level,is the proportion different? Explain.

Exhibit 9-4.A school teacher is worried that the concentration of dangerous,cancer-causing radon gas in her classroom is greater than the safe level of 4pCi/L.The school samples the air for 36 days and finds an average concentration of 4.4pCi/L with a standard deviation of 1pCi/L. Refer to Exhibit 9-4.The value of the test statistic is ___________.

Consider the following hypotheses that relate to the medical field: In this instance,a Type I error is often referred to as ___________.