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Chapter 5 and Chapter 6 Review READ: Here are some problems I selected for the review. This does not mean that only the concepts or type of problems on this review will only be on the test. Make sure you go over your homework and study the PowerPoints and notes. Key Terms Interval Estimator, Normal (z) statistic, Point Estimator, Sampling error, T-statistic, Degrees of Freedom (df), Target Parameter, Confidence interval, Confidence Level, Rejection Region, Alternative hypothesis, Null Hypothesis, Type I error, Type II error, Two-tailed Hypothesis, One-Tailed Hypothesis, Observed significance, Conclusion Understanding the Principles 1. Explain the difference between a one-tailed and a two-tailed test. 2. What is the difference between Type I and Type II errors in hypothesis testing? How does ๐ผ and ๐ฝ relate to Type I and Type II error? 3. List the four possible results of the combinations of decisions and true states of nature for a test of hypothesis. 4. Interpret the phrases โ95% confidentโ in the following statement: โWe are 95% confident that the proportion of all PCs with a computer virus falls between .12 and .18.โ Practice Problems 1. Let ๐ก0 represent a particular value of t from Table IV. Find the table values such that the following statements are true: a. ๐(๐ก โฅ ๐ก0 ) = .005, where df = 14 b. ๐(๐ก โค โ๐ก0 ๐๐ ๐ก โฅ ๐ก0 ) = .10, where df = 6 2. A random sample of 225 measurements is selected from a population, and the sample mean and standard deviation are ๐ฅฬ = 32.5 and s = 30.0, respectively. a. Use a 99% confidence interval to estimate the mean of the population, ๐. b. How large a sample would be needed to estimate ๐ to within .5 with 99% confidence? c. What is meant by the phrases โ99% confidenceโ as it is mused in this exercise? 3. According to a Food and Drug Administration (FDA) study, a cup of coffee contains an average of 115 milligrams (mg) of caffeine, with the amount per cup ranging from 60 to 180 mg. Suppose you want to repeat the FDA experiment in order to obtain an estimate of the mean caffeine content in a cup of coffee correct to within 5 mg with 95% confidence. How many cups of coffee would have to be included in your sample? 4. Find ๐ง๐ผ/2 for each of the following. a. ๐ผ = .01 b. ๐ผ = .30 5. The mean and standard deviation of a random sample of n measurements are equal to 33.9 and 3.3, respectively. a. Find a 95% confident interval for µ if n = 100. b. Find a 95% confident interval for µ if n = 400. c. Find the widths of the confidence intervals you calculated in parts a and b. What is the effect on the width of a confidence interval of quadrupling the sample size while holding the confidence coefficient fixed? 6. Strategic placement of lobster traps is one of the keys for a successful lobster fisherman. An observational study of teams fishing for the red spiny lobster in Baja California Sur, Mexico, was conducted and the results published in Bulletin of Marine Science (April, 2010). One of the variables of interest was the average distance separating trapsโcalled trap spacing โ deployed by the same team of fisherman. Trap spacing measurement (in meters) for a sample of seven teams of red spiny lobster fisherman are shown below in the table. Let ๐ represent the average of the trap spacing measurements for the population of red spiny lobster fishermen fishing in Baja California Sur, Mexico. Suppose you want to determine if the true value of ๐ differs from 95 meters. 93 99 105 94 82 70 86 a. b. c. d. Specify the null and alternative hypothesis for this test. Computer the value of the test statistic. Find the approximate p-value of the test. Select a value of ๐ผ, the probability of a Type I error. Interpret this value in the words of the problem. e. Give the appropriate conclusion, based on the results of parts c and d. 7. A key statistic used by football coaches to evaluate players is a playerโs 40-yard sprint time. Can a drill be developed for improving a playerโs speed in the sprint? Researchers at Northern Kentucky University designed and tested a speed-training program for junior varsity and varsity high school football players (The Sport Journal, Winter 2004). The training program included 50yard sprints run at a varying speeds, high knee running sprints, butt kick sprints, โcrazy legsโ straddle runs, quick fee drills, jumping, power skipping and all-out sprinting. Each in a sample of 38 high school athletes was timed in a 40-yard sprint prior to the start of the training program and timed again after completing the program. The decreases in times (measured in seconds) are listed in the table below. [Note: A negative decrease implies that the athleteโs time after completion of the program was higher than his time prior to training.] The goal of the research is to demonstrate that the training program is effective in improving 40-yard sprint times. -.01 .32 .15 .01 .1 .34 .3 .9 .1 .3 .02 .34 .24 .09 .12 .38 .25 .05 .14 .44 .05 0 .1 .08 .28 .04 .08 0 .25 .17 .5 0 .2 0 .36 .14 .21 .1 a. Find a 95% confidence interval for the true mean decrease in sprint times for the population of all football players who participate in the speed-training program. b. Based on the confidence interval, is the training program really effective in improving the mean 40-yard sprint time of high school football players? Explain. 8. Let ๐ก0 be a particular value of t. Use Table IV to find ๐ก0 values such that the following statements are true: a. ๐(โ๐ก0 < ๐ก < ๐ก0 ) = .95, where df = 16 b. ๐(๐ก โค โ๐ก0 ๐๐ ๐ก โฅ ๐ก0 ) = .10, where df = 12 9. Three-way catalytic converters have been installed in new vehicles in order to reduce pollutants from motor vehicle exhaust emissions. However, these converters unintentionally increase the level of ammonia in the air. Environmental Science & Technology (Sept. 1, 2000) published a study on the ammonia levels near the exit ramp of a San Francisco highway tunnel. The ammonia concentration (parts per million) data for eight randomly selected days during the afternoon drive time are reproduced in the table below. Find a 99% confidence interval for the population mean daily ammonia level in air in the tunnel. Interpret your result. 1.53 1.50 1.37 1.51 1.55 1.42 1.41 1.48 10. โThe Training Gameโ is an activity used in psychology in which one person shapes an arbitrary behavior by selectively reinforcing the movements of another person. Students at Georgia Institute of Technology played the game at Zoo Atlanta while participating in an experimental psychology laboratory in which they assisted in the training of animals (Teaching of Psychology, May 1998). A sample of 15 psychology students rated โThe Training Gameโ as a โgreatโ method of understanding the animalโs perspective during training of 7-1 point scale (where 1 = strongly disagree and 7 = strongly agree). The mean response was 5.87, with a standard deviation of 1.51. a. Construct a 95% confidence interval for the true mean response of the students. b. Suppose you want to reduce the width of the 95% confidence interval to half the size obtained in part a. How many students are required in the sample in order to obtain the desired confidence interval width? 11. A random sample of 175 measurements possessed a mean of ๐ฅฬ = 8.2 and standard deviation of s = .79. a. Form a 95% confidence interval for ๐. b. Test ๐ป๐ : ๐ = 8.3 against ๐ป๐ : ๐ โ 8.3. Use ๐ผ = .05. 12. Medical tests have been developed to detect many serious diseases. A medical test is designed to minimize the probability that it will produce a โfalse positiveโ or a โfalse negative.โ A false positive is a positive test result for an individual who does not have the disease, where as a false negative is a negative test result for an individual who does have the disease. a. If we treat a medical test for a disease as a statistical test of hypothesis, what are the null and alternative hypotheses for the medical test? b. What are the Type I and Type II errors for the test? Relate each to false positives and false negatives. c. Which of these errors has graver consequences? Considering this error, is it more important to minimize ๐ผ or ๐ฝ? Explain. 13. Refer to the Journal of Forest Engineering (July 1999) study on minimizing tractor skidding distances along a new road in a European forest. The skidding distances (in meters) were measured at 20 randomly selected road sites. The data is represented in the table below. The logger working on the road claims that the mean skidding distance is at least 425 meters. Is there sufficient evidence to refute this claim? Use ๐ผ = .10. Stat the appropriate conclusion in the words of the problem. 488 385 350 295 457 184 199 261 285 273 409 400 435 311 574 312 439 141 546 425 14. Health care workers who use latex gloves with glove powder on a daily basis are particularly susceptible to developing a latex allergy. Symptoms of a latex allergy include conjunctivitis, hand eczema, nasal congestion, a skin rash, and shortness of breath. Each in a sample of 46 hospital employees who were diagnosed with latex allergy based on a skin-prick test reported on their exposure to latex gloves (Current Allergy & Clinical Immunology, March 2004). The number of latex gloves used per week by the sampled workers is summarized as follows: ๐ฅฬ = 19.3 and s = 11.9. Let ๐ represent the mean number of latex gloves used per week by all hospital employees. Consider testing ๐ป0 : ๐ = 20 against ๐ป๐ : ๐ < 20. a. Give the rejection region for the test at a significance level of ๐ผ = .01. b. Calculate the value of the test statistic. c. Use the results from parts a and b to draw the appropriate conclusion. d. Compute the p-value of the test. e. Compare the p-value with ๐ผ = .01and make the appropriate conclusion.