Significance Test Review

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Walter Weaver
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1 Significance Test Review 1) Kmart brand 60W light bulbs state on the package, Average life of 1000 hours. Let µ denote the true mean life of Kmart 60W light bulbs. People who purchase this brand would be unhappy if the lifetimes of their bulbs are actually less than the advertised value. Suppose that a random sample of 100 Kmart light bulbs is selected and the lifetime for each bulb is recorded. The average lifetime of a bulb in the 100 bulb sample is 985 hours. Assume that the standard deviation for Kmart light bulbs is σ = 50. Is there significant evidence at the 5% level that Kmart light bulbs work less than 1000 hours? Follow the inference toolbox. Describe a type I error for this situation: Describe a type II error for this situation:

2 2) According to a survey of 1000 adult Americans conducted by Opinion Research Corporation, 210 of those surveyed said playing the lottery would be the most practical way for them to accumulate $200,000 in net wealth in their lifetime. Is there convincing evidence at the 5% level that more than 20% of Americans believe that playing the lottery is the best way for accumulating $200,000 in net wealth? Follow the inference toolbox. Describe a Type I Error: Describe a Type II Error:

3 3) A growing concern of employers is time spent in activities like surfing the Internet and ing friends during work hours. The San Luis Obispo Tribune summarized the findings from a survey of a large number of workers in an article that ran under the headline Who Goofs Off 2 Hours a Day? Most Workers, Survey Says (August 3, 2006). Suppose that the CEO of a large company wants to determine whether the average amount of wasted time during an 8-hour work day for employees at her company is less than the reported 120 minutes. Each person in a random sample of 10 employees was contracted and asked about daily wasted time at work. (Participants were guaranteed anonymity to ensure accurate results). The results were as follows: Do these data provide evidence that the mean wasted time for this company is less than 120 minutes? Follow the inference toolbox with a significance level of.05.

4 4) In an AP-AOL sports poll (Associated Press, December 18, 2005), 272 of 394 randomly selected baseball fans stated that they thought the designated hitter rule should either be expanded to both baseball leagues or eliminated. Based on the given information, is there sufficient evidence to conclude that a majority of baseball fans feel this way? Follow the inference toolbox and use an alpha of.01.

5 5) The article Well-Fed Crickets Bowl Maidens Over (Natural Science Update, February 11, 1999) reported that female field crickets are attracted to males that have high chirp rates and hypothesized that chirp rate is related to nutritional status. The usual chirp rate for male field crickets was reported to vary around a mean of 60 chirps per second. To investigate whether chirp rate was related to nutritional status, investigators fed male crickets a high protein diet for 8 days, after which chirp rate was measured. The mean chirp rate for crickets on the high protein diet was reported to be 109 chirps per second. Is this convincing evidence at the 1% level that the mean chirp rate for crickets on a high protein diet is greater than 60? The sample used 32 crickets and the standard deviation of the sample was 40. Follow the inference toolbox. Describe a type I error: Describe a type II error:

6 6) A report on physical characteristics of Division I starting football players in the 1988 football season revealed that the mean weight for the starting players was 105 kg. A random sample of 33 starting players from Division I teams was taken and the sample mean was found to be kg. Assume σ = 16.3kg. Is there significant evidence that the mean weight for starting football players is different from the revealed weight of 105 kg? Follow the inference toolbox and use α =.05. Construct a 95% confidence interval for the above problem. How do the interval and the test tell you the same information?

7 7) Ultrasound is often used in the treatment of soft tissue injuries. In an experiment to investigate the effect of an ultrasound and stretch therapy on knee extension, range of motion was measured both before and after treatment for a sample of physical therapy patients. A subset of the data appearing in the paper Location of Ultrasound Does Not Enhance Range of Motion Benefits of Ultrasound and Stretch Treatment is given in the table below: Subject Pre-Treatment Post-Treatment Is there evidence at the 5% level that the ultrasound and stretch treatment increases range of motion?

8 8) Duct tape to remove warts? Some people believe you can fix anything with duct tape. Even so, many were skeptical when researchers announced that duct tape may be more effective and less painful alternative to liquid nitrogen, which doctors routinely use to freeze warts. The article What a Fix-It: Duct Tape Can Remove Warts described a study conducted at Madigan Army Medical Center. Patients with warts were randomly assigned to either the duct tape treatment or the more traditional freezing treatment. Those with the duct tape wore duct tape over their wart for 6 days and then removed the tape, soaked the area in water and used an emery board to scrape the area. This process was repeated for a maximum of 2 months or until the wart was gone. The following data is from the experiment: Treatment n Success Liquid nitrogen freezing Duct tape Do the data suggest, at the 1% level, that freezing is less successful than duct tape in removing warts? Follow the inference toobox.

9 9) Reduced heart rate variability (HRV) is known to be a predictor of mortality after a heart attack. One measure of HRV is the average of normal-to-normal beat interval (in milliseconds) for a 24-hour time period. Twenty-two heart attack patients who were dog owners and 80 heart attack patients who did not own a dog participated in a study of the effect of pet ownership on HRV, resulting in the summary statistics below: HRV measure Mean Standard Deviation Owns dog Does not own dog Perform a significance test to determine if there is any difference in HRV levels for heart attack patients owning a dog vs. those not owning a dog at the 5% level. Follow the inference toolbox.

10 10) A random sample of n 1 = 378 hotel guests was taken 1 year ago, and it was found that 194 requested nonsmoking rooms. Recently, a random sample of n = hotel guests showed that 320 requested nonsmoking rooms. Do these data indicate that the proportion of hotel guests requesting a nonsmoking room has increased? Use a 1% significance level and follow the inference toolbox.

11 11) The FBI wants to study crime rates. They gathered a random sample of 10 regions in New England that gave the following violent crime rates (violent crimes per million people): Another random sample of 12 regions in the Rocky Mountain states gave the following violent crime rates (per million people): At the 1% level of significance, do the data suggest that the violent crime rate in the Rocky Mountains region is higher (per million people) than the New England region?

12 12) In the following data, B represents birth rate and D represents death rate per 1000 resident population. The data are paired by counties in the Midwest. A random sample of 16 counties gave the following information: B D B D (Reference: County and City Data Book, US Department of Commerce). Do the data indicate a difference (either way) between population average birth rate and death rate in this region? Use a 1% significance level and follow the inference toolbox.

13 For the following, state the null and alternative hypotheses in symbols and words. 1) In a report prepared by the Economic Research Department of a major bank the Department manager maintains that the average family income for a given city is $48,432. What do you conclude about the validity of the report if a random sample of 400 families shows an average income of $48,574 with a standard deviation of $2000? 2) There are two independent samples of voters in Ohio and Texas that were asked, Did you vote for John McCain? We want to test whether one or the other of these states showed significantly more support for the McCain candidacy. 3) According to Mars Candy Company, each bag of M&Ms is supposed to contain 13% red candies. My friend Ron is convinced that he is being short changed so he gathers all the candies from 600 bags of regular M&Ms, and finds that 12% are red. 4) A math test is designed to have normally distributed scores with a mean score of 75. A sample of exams scores is taken to determine whether the students are doing better or worse than expected.

14 5) According to the US Bureau of Labor Statistics, the average weekly earnings of a production worker in 1997 were $ Suppose a labor researcher wants to test to determine whether this figure is still accurate today. The researcher randomly selects 54 production workers from across the US and obtains a representative earnings statement for one week from each. The resulting sample average is $ with a standard deviation of $ ) A group of students run an experiment and find that the average percent of people who read the advertisements in a magazine is 45%. You are skeptical of this claim and think these students might be overstating the percentage. 7) The average highway gas mileage for a Honda CRV is 27 mpg. A consumer groups takes a random sample of 200 cars and records their average mpg for highway driving for a given month. They find the average to be 25.2 mpg with a standard deviation of 2 mpg. What are 4 ways to increase the power of a test? Increase the sample size Increase the alpha level Choose a sample mean further from Decrease the standard deviation µ o

Math 230 Exam 2 Name April 8, 1999

Math 230 Exam 2 Name April 8, 1999 Instructions: Answer each question to the best of your ability. Most questions require that you give a concluding or summary statement. These statements should be complete