Statistics Chapter 9 - Part B Binomial Probability of Success Name: Set: Inferences About the Binomial Probability of Success. The binomial parameter of success is defined as the number of successes divided by the number of trails; x / n. The mean of the binomial is the number of trials multiplied by the probability of success; n * p. The probability of success and the probability of failure must add up to ; p + q =. The standard deviation of the binomial is the square root of the number of trials multiplied by the probability of success multiplied by the probability of failure; sqrt (n * p * q). The binomial distribution is considered to be approximately normal if; n > 20 and if both n * p & n * q > 5. It is easier to work with the distribution of p' (probability) than with x (number of successes). Remember; p' is the observed probability, while p is the calculated probability. So we have new equations for mean and standard deviation. (I think we only have about a dozen so far.) mu of p' = p (the observed mean of the sampling distribution will be equal to the calculated mean.) sigma of p' = sqrt[(pq)/n] Question.) Forty-eight of the 57 elements in a random sample are classified as "success." What is the value of x? What is the value of n? Determine the value of p'. (Rounded to 4 decimal places.) What is the value of q'? (Rounded to 4 decimal places.) e.) Calculate the standard error for this distribution. (Correct to 3 decimal places.) Page
To build a confidence interval. (Note that this calculation uses the normal (z) distribution, not the binomial distribution.) p' - z(alpha/2) * sqrt[(p' * q')/n] p' + z(alpha/2) * sqrt[(p' * q')/n] Question 2.) Exercise 9.65 "You say tomato; burger lovers say ketchup!" According to a recent Burger Haven random survey of 803 Canadians, who were sampled independently, roughly 48% say that ketchup is their preferred burger condiment. To decide if this survey of 803 Canadians fits the properties of a binomial experiment, we must be able to identify n, p and x. What is the point estimate for the proportion of all Canadians who prefer ketchup on their burgers? (This point estimate is a probability, correct to 3 decimal places, not a whole number.) This point estimate is an estimate of a(n) ("a") Statistic ("b") Parameter ("c") Average ("d") Sample Are the conditions for using the standard normal distribution to approximate the binomial satisfied? ("a") No, n < 20 ("b") No, n * p < 5 ("c") No, n * q < 5 ("d") No, this was not a random sample. ("e") No, the trials were not independent. ("f") Yes, n>20, n*p & n*q > 5 and this is a random sample, and all trials were independent. Calculate E, the maximum error of the estimate, for a 95% confidence interval for a binomial experiment of 803 trials that results in an observed proportion of 0.48. (Correct to 3 decimal places.) Remember, inferences about the binomial probability of success use the standard normal table, not the Student's t-distribution. e.) The margin of error associated with this study would be reported as plus or minus what percent? (Rounded to decimal place, include the % sign in your answer.) f.) Find the 95% confidence interval for the true population proportion based on a binomial experiment of 803 trials that results in an observed proportion of 0.48. Use the rounded value for E, and round your lower and upper limits to 3 decimal places. Present the lower limit first, and the upper limit second. g.) Draw a diagram of the confidence interval. Page 2
Question 3.) Exercise 9.38 & 9.39 & 9.40 The marketing department of an instant-coffee company conducted a survey of married men to determine the proportion who preferred their brand. Twenty-three of the 7 men in the random sample preferred the company's brand. Use an 80% confidence interval to estimate the proportion of all married men who prefer this company's brand of instant coffee. Round your two answers to 3 decimal places. A company is drafting an advertising campaign that will involve endorsements by noted athletes. In order for the campaign to succeed, the endorser must be both highly respected and easily recognized. A random sample of 3 prospective customers are shown photos of various athletes. If the customer recognizes an athlete, then the customer is asked whether he or she respects the athlete. In the case of a top woman golfer, 9 of the 3 respondents recognized her picture and indicated that they also respected her. (This must have been the night before, not the morning after.) At the 98% level of confidence, find the confidence interval for the true proportion for which this woman golfer is both recognized and respected? Round your two answers to 3 decimals. A local auto dealership advertises that 89% of customers whose autos were serviced by their service department are pleased with the results. As a researcher, you take exception to this statement because you are aware that many people are reluctant to express dissatisfaction even if they are not pleased. A research experiment was set up in which those in the sample had received service by this dealer within the past two weeks. During the interview, the individuals were led to believe that the interviewer was new in town and was considering taking his car to this dealer's service department. Of the 58 sampled, 9 said that they were dissatisfied and would not recommend the department. Estimate the proportion of dissatisfied customers using a 98% confidence interval. Round your two answers to 3 decimals. Does the dealers claim seem reasonable? (Yes or No) Page 3
Finding the required sample size. See equation 9.8 on page 436 of your textbook. (Look for this equation on your formula sheets.) With this equation we now have three types of p to worry about. (Sounds painful, see a doctor!) The p without ' or * is the population parameter, this is what we are trying to estimate. The p' is the observed probability of success in our sample. The p* is our best guess, before we actually gather any data. Question 4.) Exercise 9.77 According to "Canada Today" 86% of all drivers use their seat belts. You wish to conduct a survey in your city to create a confidence interval for the percentage of the drivers who use seat belts. Use the national figure of 86% for your initial estimate of p. Find the sample size if you want your estimate to be within 0.02 with 95% confidence. (4 points) What effect does changing E, the maximum error of the estimate, have on the sample size? ("a") Larger maximum error - smaller sample size required. ("b") Larger maximum error - larger sample size required. ("c") Smaller maximum error - smaller sample size required. What effect does changing the level of confidence have on the sample size? ("a") Higher level of confidence - smaller sample size required. ("b") Higher level of confidence - larger sample size required. ("c") Lower level of confidence - larger sample size required. Page 4
When testing hypotheses about the binomial probability parameter, p, the Null Hypothesis and the Alternate Hypothesis must both be in terms of p instead of mu. The Null Hypothesis must still contain "=", the equals sign. Depending upon the situation, the Alternate Hypothesis can contain either "greater than", >, "less than", <, or "not equal to", <>. The value stated in the Alternate Hypothesis is always the same as the value that was stated in the Null Hypothesis. Question 5.) Exercise 9.79 For each of the following situations chose the letter which represents the Alternate Hypothesis that would be used to test these claims. (Remember the alternative hypothesis is always the negation of the claim.) The administration states that the percentage of students working is less than 60%, but the student union wants to test this, they believe that more than 60% of all students work part-time jobs during the academic year. ("a") Ha: p > 60% ("b") Ha: p < 60% ("c") Ha: p <> 60% The Las Vegas bookies think your team will lose tonight, but you think they will win. ("a") Ha: p > 50% ("b") Ha: p < 50% ("c") Ha: p <> 50% The proportion of cigarette smokers who are interested in quitting is one-third. ("a") Ha: p > 0.333% ("b") Ha: p < 0.333% ("c") Ha: p <> 0.333% At least 50% of all parents believe in spanking their children when appropriate. ("a") Ha: p > 50% ("b") Ha: p < 50% ("c") Ha: p <> 50% e.) The polls show that the school budget will not pass, but you believe a majority of voters will vote for the school budget this year. ("a") Ha: p > 50% ("b") Ha: p < 50% ("c") Ha: p <> 50% f.) At least three-quarters of the trees in our county were seriously damaged by the storm. ("a") Ha: p > 75% ("b") Ha: p < 75% ("c") Ha: p <> 75% g.) Coins should be evenly balanced, but you suspect the coin is not balanced fairly. ("a") Ha: p > 50% heads. ("b") Ha: p < 50% heads. ("c") Ha: p <> 50% heads. h.) A random number generation program should not exhibit bias, but the single digit numbers generated by the computer seem to be biased towards even numbers. ("a") Ha: p > 50% even. ("b") Ha: p < 50% even. ("c") Ha: p <> 50% even. Page 5
Hypothesis Testing z* = (p' - p) / sqrt[(pq)/n] standard error of p' = sqrt[(pq)/n] Question 6.) Exercise 9.87 An insurance company states that 90% of its claims are settled within 36 days. A consumer group selected a random sample of 64 of the company's claims to test this statement. These claims dealt with situations that were independent of each other. The consumer group found that 52 of the claims were settled within 36 days. Do they have sufficient evidence to support their contention that fewer than 90% of the claims are settled within 36 days? Use alpha = 0.02. e.) Describe the population parameter of interest. ("a") P(The company is lying.) ("b") P(The company is telling the truth.) ("c") P(The claims are settled within 36 days.) ("d") P(Mean is more than 90%.) What is the Alternate Hypothesis? ("a") Ha: mu < 0.9 ("b") Ha: p = 0.9 ("c") Ha: p > 0.9 ("d") Ha: p < 0.9 Are the assumptions satisfied? ("a") Yes, the sample size is greater than 30. ("b") Yes, the parent population is normal. ("c") Yes, the sample was randomly selected and independent. ("d") No, we do not know the standard deviation. Identify the probability distribution to be used. ("a") Binomial ("b") Standard normal ("c") Student's t-distribution ("d") Chi-square Determine the level of significance. f.) Determine z(alpha) Remember this will be negative if Ha contains <. g.) Calculate z* (Correct to 2 decimals.) Do not round off p' before using it in this equation. h.) i.) Make a decision based on the classical approach. ("c") Accept Ha Find the p-value. (Correct to 4 decimals.) j.) Make a decision based on the probability approach. ("c") Accept Ha Page 6
Question 7.) Exercise 9.88 A recent survey conducted by ZOOM and Applied Research & Consulting LLC reported that the events of September, 200, have motivated kids to volunteer and that more than 8% volunteer. A disbeliever of this information took a separate random sample of 46 kids in an attempt to show that the true percentage of kids who volunteer is less than 8%. In the sample taken by the disbeliever 34 of the kids said that they do volunteer work. Complete the following hypothesis test to decide if the disbeliever has sufficient evidence to support his or her contention that fewer than 8% of kids do volunteer work. Use alpha = 0.005. Calculate z* (Correct to 2 decimals.) Do not round off p' before using it in this equation. Draw a diagram which shows the critical region and z*. Find the p-value. (Correct to 4 decimals.) Make a decision. ("c") Accept Ha Question 8.) Exercise 9.90 The full time student body of a college is 5% men and 49% women. Consider a random sample of students (29 men, 2 women) from an introductory chemistry course. Complete the following hypothesis test to decide if the proportions of male and female students who take the chemistry course are the same as those in the whole student body. Use alpha = 0.. Calculate z* based on the proportion of men. (Correct to 2 decimals.) Do not round of p' before using it in this equation. Draw a diagram which shows the critical region and z*. Find the p-value. (Correct to 4 decimals.) (Remember, double the p-value for a 2-tail test.) Make a decision. ("c") Accept Ha Page 7
Question 9.) Exercise 9.93 USA Today ("Facing a crowd isn't easy," May 30, 2002) reported that 33% of the country's professional women fear public speaking. Suppose you conduct a survey of 832 randomly chosen professional women to test whether the true proportion is less than stated in the article. Of the 832 sampled, 265 feared public speaking. Complete the following hypothesis test to decide if the proportion of professional women in the sample is less than the proportion stated in the article. Use alpha = 0.02. Calculate z*. (Correct to 2 decimals.) Do not round off p' before using it in this equation. Draw a diagram which shows the critical region and z*. Find the p-value. (Correct to 4 decimals.) Make a decision. ("c") Accept Ha Question 0.) Exercise 9.92 The popularity of personal watercraft (PWC's also known as jet skis) continues to increase, despite the apparent danger associated with their use. In fact, a sample of 57 watercraft accidents reported to the Nebraska Game and Parks Commission in 997 revealed that 86% of them involved PWC's even though only 5% of the motorized boats registered in the state are PWC's. Suppose the national average proportion of water craft accidents in 997 involving PWC's was 79%. Does the watercraft accident rate for PWC's in Nebraska exceed the rate in the nation as a whole? Use a 0.0 level of significance Calculate z* (Correct to 2 decimals.) Do not round off p' before using it in this equation. Draw a diagram which shows the critical region and z*. Find the p-value. (Correct to 4 decimals.) Make a decision. ("c") Accept Ha Page 8
Bonus Question ) Exercise 9.42 The so-called "glass ceiling" and numerous other reasons have prevented women from reaching the top of the corporate employment ladder compared to men. Fortune magazine ("The Global Glass Ceiling," October 2, 998) reported that women make up 2% of corporate directors in the Fortune 500 companies, even though women represent a much higher percentage (40%) of the total work force employed in management positions in America. The percentage of female corporate directors and officers, however, has been rising steadily, and power appears to be shifting to people who are not in traditional corporate Canada. You wish to conduct a study to estimate the percentage of female corporate directors in the companies with headquarters in your province. Assume the population proportion for Canada is 2% as reported by Fortune magazine for America. What sample size must you use if you want your estimate to be within 0.04 with 98% confidence? (4 points) What sample size must you use if you want your estimate to be within 0.05 with 90% confidence? What sample size must you use if you want your estimate to be within 0.09 with 80% confidence? Page 9
Statistics Chapter 9 - Part B Binomial Probability of Success Answers 6/29/2009 2:26 Question.) x is the number of successes = 48 (with only two outcomes, "success' and "failure"). n is the sample size = number of independent trials = 57. p' = 0.3057 q' = 0.6943 e.) 0.037 Question 2.) Question 6.) Question 0.) 0.479 c.3 b d Ha: p < 0.9 p' = 0.86 f c z(alpha/2) = 2.33 0.035 b 0.0968 e.) 3.5% e.) 0.02 b Fail to Reject Ho f.) 0.445 f.) -2.05 0.55 g.) -2.33 Bonus Question ) h.) a Reject Ho 359 i.) 0.0099 5 j.) a Reject Ho 22 Question 7.) -2.87 z(alpha) = -2.58 Question 3.) 0.002 0.5 a Reject Ho 0.244 0.086 0.25 0.044 0.266 95.6% 73.4% Yes, the dealer's claim is reasonable. Question 4.) Question 8.) 57 0.99 a p' = 0.58 b z(alpha/2) =.65 0.3222 Question 5.) b Fail to Reject Ho a Ha: p > 60% a Ha: p > 50% Question 9.) c Ha: p <> 0.333% -0.7 b Ha: p < 50% p' = 0.385 z(alpha) = -2.05 e.) a Ha: p > 50% 0.2420 f.) b Ha: p < 75% b Fail to Reject Ho g.) c Ha: p <> 50% heads. h.) a Ha: p > 50% even.