12.2 Inference for a Population Proportion We are interested in the unknown proportion p of a population that has some outcome – call the outcome we are looking for a “success.” The statistic that estimate the parameter p is the sample proportion p-hat = count of successes in sample count of observations sample

Recall: The sampling distribution of p-hat… The mean of the sampling distribution is p. The sample proportion p-hat is an unbiased estimator of the population proportion p. The standard deviation of p-hat is

Solution to not knowing p  To test the null hypothesis that the unknown p has a specific value, replace p by in the test statistic: to get  In a confidence interval for p: will be close to p if n is large, so replace the std. dev. by the standard error of.

Assumptions for inference about a proportion The data are an SRS from the population of interest The population is at least 10 times as large as the sample. For a test of, the sample size n is so large that: and For a confidence interval, n is so large that and (successes)(failures)

Example A coin that is balanced should come up heads half the time in the long run. The population for coin tossing contains the results of tossing the coin forever. The parameter p is the probability of a head, which is the proportion of all tosses that give a head. The tosses we actually make are an SRS from this population. The French naturalist Count Buffon ( ) tossed a coin 4040 times. He got 2048 heads. The sample proportion of heads is That is more than one-half. Is this evidence that Buffon’s coin was not balanced? Find the 95% confidence interval.

Recall: n for a desired margin of error To determine the sample size n that will yield a level C confidence interval for a population proportion p, solve the following for n: is the standard normal critical value for level of confidence we want is a guess (or is from a study).

Many colleges that once enrolled only male or only female students have become coeducational. Some administrators and alumni were concerned that the academic standards of the institutions would decrease with the change. One formerly all-male college undertook a study of the first class to contain women. The class consisted of 851 students, 214 of whom were women. An examination of first-semester grades revealed that 15 of the top 30 students were female. 1)What is the proportion of women in the class? Call this value p-nought. 2)Assume that the number of females in the top 30 is approximately a binomial random variable with n=30 and unknown probability p of success. In this case success corresponds to the student being female. What is the value of p-hat? 3)Are women more likely to be top students than their proportion in the class would suggest? State hypotheses that ask this question, carry out a significance test, and report your conclusion in non- technical language.

Methods of Poll Explained The Times-Dispatch/12 News poll was conducted by the research department of Medial General, Inc., parent company of the Times-Dispatch. Based on telephone interviews October 23 through Wednesday with 502 respondents who identified themselves as registered voters, the survey had a sampling error of plus or minus 4.5 percentage points. In other words, one could say with 95 percent certainty that the results of the poll would vary 4.5 percent in either direction if the entire adult population of Virginia had been polled. Verify that the newspaper’s margin of error is correct. What is the exact margin of error?