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EXAM REVIEW Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley MODELING THE DISTRIBUTION OF SAMPLE PROPORTIONS (CONT.) A picture of what we just discussed is as follows: EXAMPLE A candy company claims that its jelly bean mix contains 45% red jelly beans. Suppose that the candies are packaged at random with 100 jelly beans each. What’s the probability that a bag will contain more than 50% red jelly beans? Slide 1- 3 PROPORTIONS: ONE-SAMPLE - SE AND Z* HYPOTHESIS TESTING The conditions for the one-proportion z-test are the same as for the one proportion z-interval. We test the hypothesis H 0: p = p 0 using the statistic pˆ p0 z SD pˆ where SD pˆ p0 q0 n When the conditions are met and the null hypothesis is true, this statistic follows the standard Normal model, so we can use that model to obtain a P-value. Slide 1- 4 PROPORTIONS: ONE-SAMPLE CONFIDENCE INTERVALS When the conditions are met, we are ready to find the confidence interval for the population proportion, p. The confidence interval is pˆ z SE pˆ where ˆˆ SE( pˆ ) pq n The critical value, z*, depends on the particular confidence level, C, that you specify. Slide 1- 5 PROPORTIONS: ONE-SAMPLE - EXAMPLE A state university wants to increase its retention rate of 10% for graduating students from the previous year. After implementing several new programs to increase retention during the last two years, the university re-evaluated its retention rate using a random sample of 352 students. The new retention rate was 12%. Slide 1- 6 PROPORTIONS: ONE-SAMPLE - EXAMPLE Test the hypothesis that the retention rate had increased and state your conclusion with a 98% confidence interval. Also test the hypothesis with a z-test using a significance level of 0.01 Slide 1- 7 MEANS: ONE-SAMPLE – T-TESTING A practical sampling distribution model for means When the conditions are met, the standardized sample mean y t SE y follows a Student’s t-model with n – 1 degrees of freedom. s We estimate the standard error with SE y n Slide 1- 8 MEANS: ONE-SAMPLE - HYPOTHESIS One-sided alternatives Ha: μ>hypothesized value Ha: μ <hypothesized value Two-sided alternatives Ha: μ ≠ hypothesized value Slide 1- 9 MEANS: ONE-SAMPLE – CONFIDENCE INTERVALS One-sample t-interval for the mean When the conditions are met, we are ready to find the confidence interval for the population mean, μ. The confidence interval is SE y n 1 where the standard error of the mean is y t s SE y n * The critical value tn1depends on the particular confidence level, C, that you specify and on the number of degrees of freedom, n – 1, which we get from the sample size. Slide 1- 10 MEANS: ONE-SAMPLE – EXAMPLE A sociologist develops a test to measure attitudes about public transportation, and 50 randomly selected subjects are given the test. Their mean score is 85 and their standard deviation is 15. Construct a 95% confidence interval for the mean score of all such subjects. Slide 1- 11 PROPORTIONS: TWO-SAMPLE - HYPOTHESIS The typical hypothesis test for the difference in two proportions is the one of no difference. In symbols, H0: p1 – p2 = 0. The alternatives: Ha: p1 –p2 > 0 Ha: p1 –p2 < 0 Ha: p1 –p2 ≠ 0 Slide 1- 12 PROPORTIONS: TWO-SAMPLE - SE AND Z* HYPOTHESIS TESTING We use the pooled value to estimate the standard error: pˆ pooled qˆ pooled pˆ pooled qˆ pooled SE pooled pˆ1 pˆ 2 n1 n2 Now we find the test statistic: pˆ1 pˆ 2 0 z SE pooled pˆ1 pˆ 2 When the conditions are met and the null hypothesis is true, this statistic follows the standard Normal model, so we can use that model to obtain a P-value. Slide 1- 13 CALCULATING THE POOLED PROPORTION The pooled proportion is pˆ pooled where Success1 Success2 n1 n2 Success1 n1 pˆ1 and Success2 n2 pˆ 2 If the numbers of successes are not whole numbers, round them first. (This is the only time you should round values in the middle of a calculation.) Slide 1- 14 PROPORTIONS: TWO-SAMPLE CONFIDENCE INTERVALS When the conditions are met, we are ready to find the confidence interval for the difference of two proportions: The confidence interval is pˆ1 pˆ 2 z SE pˆ1 pˆ 2 where SE pˆ1 pˆ 2 pˆ1qˆ1 pˆ 2 qˆ2 n1 n2 The critical value z* depends on the particular confidence level, C, that you specify. Slide 1- 15 PROPORTIONS: TWO-SAMPLE - EXAMPLE A survey of randomly selected college students found that 50 of the 100 freshman and 60 of the 125 sophomores surveyed had purchased used textbooks in the past year. Test for a difference between the two student groups using a significance level of 0.05. Slide 1- 16 MEANS: TWO-SAMPLE - HYPOTHESIS One-sided alternatives Ha: μ1 – μ2 >0 Ha: μ1 – μ2 <0 Two-sided alternatives Ha: μ1 – μ2 ≠ 0 Slide 1- 17 MEANS: TWO-SAMPLE – T-TESTING When the conditions are met, the standardized sample difference between the means of two independent groups y1 y2 1 2 t SE y1 y2 can be modeled by a Student’s t-model with a number of degrees of freedom found with a special formula. We estimate the standard error with s12 s22 SE y1 y2 n1 n2 MEANS: TWO-SAMPLE – DEGREES OF FREEDOM The special formula for the degrees of freedom for our t critical value is a bear: 2 s12 s22 n1 n2 df 2 2 1 s12 1 s22 n1 1 n1 n2 1 n2 Because of this, we will let technology calculate degrees of freedom for us! Slide 1- 19 MEANS: TWO-SAMPLE – CONFIDENCE INTERVAL When the conditions are met, we are ready to find the confidence interval for the difference between means of two independent groups. The confidence interval is y1 y2 t df SE y1 y2 where the standard error of the difference of the means is s12 s22 SE y1 y2 n1 n2 The critical value depends on the particular confidence level, C, that you specify and on the number of degrees of freedom, which we get from the Slide sample sizes and a special formula. 1- 20 MEANS: TWO-SAMPLE – EXAMPLE Two types of cereal brands are being tested for sugar content Brand Yummy – n=100, Ӯ=5, s=2 Brand Yuck – n=150, Ӯ=4.5, s=2 Construct a 95% confidence interval for the difference between the two brands. Slide 1- 21 UPCOMING IN CLASS Exam #2 Wednesday Data Project Due by 5pm Thursday December 5th via email or my department mailbox. Finals (optional) Wednesday December 11th 1-3pm