AP Statistics Ch 12 Inference for Proportions

Transcription

1 Ch 12.1 Inference for a Population Proportion Conditions for Inference The statistic that estimates the parameter p (population proportion) is the sample proportion p ˆ. p ˆ = Count of successes in the sample = X Count of observations in the sample n The data must be an SRS from the population of interest and the population must be at least 10 times as large as the sample. For a test of H 0 : p = p 0, the sample size is so large that np o 10 and n(1-p o ) 10. For a confidence interval, the sample size is so large that nˆ p 10 and n(1 p ˆ ) 10. Example Tuition Survey o Glenn wonders what proportion of the students at his school think that tuition is too high. He interviews an SRS of 50 of the 2400 students at his college. Thirty-eight of those interviewed think tuition is too high. o Describe the population parameter p and the statistic p ˆ. o Determine if the conditions are met for a confidence interval. The z-procedure Draw an SRS of size n from a large population with unknown proportion p of successes. An approximate level C confidence interval for p is p ˆ (1 p ˆ ) p ˆ ± z * n To test the hypothesis H 0 : p = p 0, compute the z statistic p ˆ p z = 0 p 0 (1 p 0 ) n with P-values calculated from the standard normal distribution. page 1

2 Example We Want to be Rich o In a recent year, 73% of first-year college students responding to a national survey identified being very well-off financially as an important personal goal. A state university finds that 132 of an SRS of 200 of its first-year students say that this goal is important. o Give a 95% confidence interval for the proportion of all first-year students at the university who would identify being well-off as an important personal goal. o Is there good evidence that the proportion of all first year students at this university who think being well-off is important differs from the national value, 73%? page 2

3 Choosing the Sample Size Use the following to calculate the sample size needed to obtain a given margin of error. Use p * = 0.5 for a conservative estimate. p * (1 p * ) z * m n If you suspect p* to be close to 0 or 1, the sample size will be much larger than you need, so you can use a better guess from a pilot study. Example Starting a Night Club o A college student organization wants to start a nightclub for students under the age of 21. To assess support for this proposal, they will select an SRS of students and ask each respondent if he or she would patronize the type of establishment. They expect that about 70% of the student body would respond favorably. o What sample size is required to obtain a 90% confidence interval with an approximate margin of error of 0.04? o Suppose that 50% of the sample responds favorably. Calculate the margin of error of the 90% confidence interval. page 3

4 Ch 12.2 Comparing Two Proportions The Sampling Distribution of ˆ p 1 ˆ p 2 In a two-sample problem, we want to compare two populations or the responses to two treatments based on two independent samples. We compare populations by doing inference about the difference p 1 p 2 between the population proportions. The statistic that estimates this difference is the difference between the two sample proportions, p ˆ 1 p ˆ 2. When calculating the standard deviation for the difference in sample proportions, remember that it is the variances that add. Confidence Intervals for p 1 p 2 An approximate level C confidence interval for comparing two proportions is p ˆ ( p ˆ 1 p ˆ 2 ) ± z * 1 (1 p ˆ 1 ) p + ˆ 2 (1 p ˆ 2 ) n 1 n 2 Conditions: o Populations must be at least 10 times as large as the samples o Independent random samples o n 1 p ˆ 1 5, n 1 (1 p ˆ 1 ) 5, n 2 p ˆ 2 5, and n 2 (1 p ˆ 2 ) 5 page 1

5 Example Free Speech o The 1958 Detroit Area Study was an important investigation of the influence of religion on everyday life. The sample was basically a simple random sample of the population of the metropolitan area of Detroit, Michigan. Of the 656 respondents, 267 were white Protestants and 230 were white Catholics. The study took place at the height of the Cold War. One question asked if the right of free speech included the right to make speeches in favor of communism. Of the 267 white Protestants, 104 said yes, while 75 of the 230 white Catholics said yes. Give a 95% confidence interval for the difference between the proportion of Protestants who agreed that communist speeches are protected and the proportion of Catholics who held this opinion. page 2

6 Significance Tests for ˆ p 1 ˆ p 2 Significance tests of H 0 : p 1 = p 2, use the pooled sample proportion p ˆ = Count of successes in both samples combined = X 1 + X 2 Count of observations in both samples combined n 1 + n 2 To test the hypothesis H 0 : p 1 = p 2 first find the pooled proportion p ˆ of successes in both samples combined. Then compute the z statistic p ˆ z = 1 p ˆ 2 p ˆ (1 p ˆ ) n 1 n 2 with P-values calculated from the standard normal distribution. Conditions: o Populations must be at least 10 times as large as the samples o Independent random samples o n 1 p ˆ 1 5, n 1 (1 p ˆ 1 ) 5, n 2 p ˆ 2 5, and n 2 (1 p ˆ 2 ) 5. Example Preventing Strokes o Asprin prevents blood from clotting and so helps prevent strokes. The Second European Stroke Prevention Study asked whether adding another anticlotting drug named dipyridamole would be more effective for patients who had already had a stroke. Here are the data on strokes and deaths during the two years of the study: Treatment # of Patients # of Strokes # of Deaths Asprin alone Aprin + dipyridamole o The study was a randomized comparative experiment. Outline the design of the study. page 3

7 o Is there a significant difference in the proportion of strokes in the two groups? o Is there a significant difference in the death rates for the two groups? page 4

8 AP Statistics Ch 10 Introduction to Inference Ch 10.4 Inference as Decision Type I and Type II Errors Type I error: Rejecting a null hypothesis when it is true. The probability of a Type I error is α, the significance level of the test. Type II error: Accepting a null hypothesis when it is false. Power Power of a test: Probability of correctly rejecting a null hypothesis. Power is 1 minus the probability of a Type II error. Calculations of power are done to check the sensitivity of a test. Using a significance test with low power makes it unlikely that you will find a significant effect even if the truth is far from the null hypothesis. You can increase the power of a test by increasing alpha or by increasing the sample size. page 1

9 AP Statistics Ch 10 Introduction to Inference Example Water samples o Water samples are taken from water used for cooling as it is being discharged from a power plant into a river. It has been determined that as long as the mean water temperature of the discharged water is at most 150 F, there will be no negative effects on the river s ecosystem. To investigate whether the plant is in regulations that prohibit a mean discharge-water temperature above 150, 50 water samples will be taken at randomly selected times, and the temperature of each sample recorded. Your hypotheses will be tested at the 5% level. o State your null and alternative hypotheses. o Describe a Type I error in the context of this problem. What is the probability of making a Type I error? o Describe a Type II error in the context of this problem. Give two ways to reduce the probability of a Type II error (and thus raise the power). o What type of error would you consider more serious? Explain. page 2