Chapter 7 EXERCISES A standard deck of playing cards has 52 cards, so the probability of drawing any particular card is 1/52. In five consecutive draws (without replacement) from a standard deck of playing cards, what is the probability of obtaining a royal straight flush (10, jack, queen, king, ace) in hearts? 3. Toss a coin, roll one die, and draw one card from a standard deck. What is the probability of getting the following combination: tails, 4, king? 4. What is the probability of drawing either an ace or a king on any single draw? What is the probability of drawing either a heart or a spade? What is the proba- bility of drawing an ace and a king on two consecutive draws? 5. What is the probability of drawing aces on four consecutive draws without replacement? 6. The probability of exposure to a virus in a blood transfusion is .001 at a par- ticular hospital. A man who had a blood transfusion at that hospital is later tested for the presence of the virus, and the test is negative. The probability that the test will result in a false negative (i.e., that it will fail to detect the presence of the virus) is .01. What is the probability that the man was exposed to the virus and that the screening test failed to detect it? 7, Harry has just taken a 20-question true-false test for which he has not studied. Assuming random guessing, what is the probability that Harry will miss all 20 questions? 9. You have just bought a ticket for the Chamber of Commerce lottery, and your friend has bought two tickets. You read in the local paper that 500 tickets were sold, and the winner will be chosen tomorrow. a. What is the probability that you will win? 1500 1500 b. What is the probability that your friend will win? C. What is the probability that either you or your friend will win? d. What is the probability that you and your friend will win? 10. Assume that a lottery was conducted in a small office and only 10 tickets were sold; you and a friend each bought 1 ticket. Two winners will be selected. What is the probability that your friend will get the first prize and you will get the second? What is the probability that you will get the first prize and your friend will get the second? 11 Given the situation in Exercise 10, what is the probability that you two will receive both prizes? 12. What is the probability of three heads in five flips of a fair coin? Of four heads? 13: A study was made of outcomes from psychotherapy and education-whether the client had a college degree. The results of the study are summarized in the following table of probabilities: 1. What are the characteristics of the normal curve? How does the standard nor- mal curve differ from other normal curves? 3. of We 2. know that a reading test has a mean (X) of 10 and standard deviation (s) Using the normal distribution, answer the following a questions: a. What is the percentile rank corresponding to a score of 11.2? b. If we tested 1,000 randomly selected students, how many would be ex- pected to score 5 or lower? c. What reading score corresponds to a percentile rank of 75%?30%? d. What is the probability of a score of 13 or higher? 5. In an introductory psychology class, 83 students have taken a sociability rating scale. The possible ratings are from 1 to 7 with a low score indicating low sociability and a high score indicating high sociability. The average rating has been 4.23, with S = 1.02. Using the normal curve table, answer the following: a. What is the percentile rank of a student with a rating of 3.11? b. How many of the 83 students had scores between 4.88 and 5.62? c. What scores are so deviant that they occurred 1% or less of the time? d. How many students had scores of 2.55 or less? 6.11 or more? e. What is the score at the 83rd percentile? f. What is the probability of a score of 3.5 or less? In an introductory English class, 100 students were given 20 words to define. The data from this test were approximately as follows: X = 1,000, EX2 = 10,400. Answer the following: a. What is the percentile rank of a person defining 9 words correctly? b. How many people scored above 13? c. What percentage of the class scored between 7 and 11? d. What scores are so deviant that less than 7% of the class made them? 9. On the final exam in a large statistics class, the average score was 76 with a standard deviation of 7. Assuming a normal population, what scores were so deviant that their probability of occurrence was .01 or less? What was the probability of a score of 60 or less? 10. Data collected by the highway patrol indicate that the average speed on a stretch of interstate is 63 mph, with a standard deviation of 15. What is the likelihood of observing a vehicle traveling 80 mph or more? 45 mph or less? If a police officer observed 953 vehicles in an afternoon, how many of them (rounded to the nearest whole vehicle) would be traveling 90 mph or more? 11. A faculty member collected data on the length of faculty meetings in her de- partment over a 3-year period. The average was 110 minutes, with a standard deviation of 25 minutes. Over the next 40 meetings, how many would-) be ex- pected to last 1 hour or less? If faculty members start leaving when a meeting runs more than 2.5 hours, what is the probability of a walk-out at the next meeting? What meeting lengths are so extreme that they are likely to occur 5% of the time or less? EXERCISES 1 a. Given X = 100, S = 20, and N = 25, find the 99% CI for u. b. IfN = 147, what is the 95% CI for u? Note that 146 is closer to 120 than it is to 00. 2. Find the 99% and 95% CIs for each of the following samples: a. X = 15, N = 17,s = 5 b. X = 240, N = 40, xx = 1.6 c. N = 170, X = 1,445, X² = 12,343.25 3, Find the 95% and 99% CIs for u for the following scores: X f X f 42 1 22 2 30 2 21 1 26 6 19 2 25 1 18 4 24 1 17 2 23 3 7 Briefly define the following terms: a. power of a test b. a error C. B error d. directional hypothesis 14. Over a 20-year period, the average grade point average of applicants to graduate program has been found to be 3.27. This year the mean grade our average of the 57 students applying to our graduate program is 3.53, with point standard deviation of 0.29. Test the hypothesis that the current sample came a from a population in which u = 3.27. Use an a level of .05 and a nondirec- tional test. 15. Use the sample data from Exercise 14 to compute the 95% and 99% confidence intervals for u. Is u = 3.27 in the 95% CI? Did you reject Ho in Exercise 14? (If the null hypothesis value of u is not in the 95% CI, then you will reject Ho with p < .05.) Chapter 9 2nd Part USING SPSS-EXAMPLE AND EXERCISE SPSS has a specific, easy-to-use procedure for computing the one-sample t test. Example: We will use SPSS to work Self-Test Exercise 7. The steps are as follows: 1 Start SPSS, enter the data, and name the variable hypochon. 2 Analyze> Compare Means>One-Sample T Test. 3 Move hypochon into the Test Variables box and enter 49.2 as the Test Value. 4 Note that the Test Value is the hypothesized value for for the null hypothesis. 5 Click OK and the solution should appear in the output Viewer window. 6 To obtain the 95% CI for u, we must trick SPSS a bit and enter a Test Value of 0 and click OK. 7 Only the CI values should be read from this portion of the output. Notes on Reading the Output 1 1. The column labeled "Sig. (2-tailed)" gives the exact p value for the computed t = 2.425. This means that p = .034, and we need not look up the critical values for t at the .05 or .01 levels. Our rule for rejecting Ho can now be based on whether "Sig. (2-tailed)" or p .05. 22. The 95% CI is the CI on the difference between the sample mean and the hypothesized mean. In order to obtain the correct CI, we must re-run the analysis with a Test Value set to 0. This portion of the output will give the correct CI, but the t value will not be correct and should be ignored. The solution output for the data of Self-Test Exercise 7 is as follows: T-TEST /TESTVAL=49.2 /MISSING=ANALYSIS /VARIABLES=hypochor /CRITERIA=CIN (.95) . T-Test One-Sample Statistics Std. Std. Error N Mean Deviation Mean HYPOCHON 12 58.4167 13.1665 3.8008 Sig. (2-tailed) is the exact p value for the computed t= 2.425 and p=.034. One-Sample Test Test Value = 49.2 95% Confidence Interval of the Sig. Mean Difference t df (2-tailed) Difference Lower Upper HYPOCHON 2.425 11 .034 9.2167 .8511 17.5822 T-TEST /TESTVAL=0 /MISSING=ANALYSIS /VARIABLES=hypochon /CRITERIA=CIN (.95) T-Test One-Sample Statistics Std. Std. Error N Mean Deviation Mean HYPOCHON 12 58.4167 13.1665 3.8008 Only the 95% CI is correct for the following output. One-Sample Test Test Value = 0 95% Confidence Interval of the Sig. Mean Difference t df (2-tailed) Difference Lower Upper HYPOCHON 15.369 11 .000 58.4167 50.0511 66.7822 Exercise Using SPSS 1We have conducted a study of the verbal skills of females. The task was to unscramble 20 sentences within a 10-minute period. (Example: free are things best the life in-The - best things in life are free.) Each of the 20 participants received a score indicating the number of sentences she unscrambled correctly. Several previous studies over the last 2 years have indicated that females averaged a score of 9.0 on the task. Using SPSS, test the hypothesis that this year's sample performed differently than in the past. Also provide a 95% CI for the population mean for this sample. The data are as follows: 15, 15, 14, 14, 13, 13, 13, 11, 11, 11, 11, 10, 10, 9, 9, 9, 8, 8, 6, 3. Write a brief conclusion for the hypothesis test. Also, interpret the confidence interval.