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How much sleep did you get last night? 1. 2. 3. 4. 5. 6. <6 6 7 8 9 >9 0% 0% 1 2 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 0% 0% 0% 0% 3 4 5 6 Slide 1- 1 Exam Review Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Possible Tests One-proportion z-test Two-proportion z-test One-sample t-test for mean Two-sample t-test for differences of means Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 3 Inferences about Proportions – One Sample Test Everyone (100%) believes in ghosts More than 10% of the population believes in ghosts Less than 2% of the population has been to jail 90% of the population wears contacts Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 4 Proportions: One-sample - Hypothesis Hypothesis Test H0: parameter = hypothesized value HA: parameter < hypothesized value HA: parameter ≠ hypothesized value HA: parameter > hypothesized value Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 5 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. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 6 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 H0: p = p0 using the statistic where SD pˆ pˆ p0 z 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. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 7 Proportions: One-sample - SE and hypothesis testing For a sample proportion, the standard error is SE pˆ ˆˆ pq n For the sample mean, the standard error is s SE y n Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 8 Critical Z* and significance level twosided α= .20 CI = 80% z*=1.282 α= .10 CI = 90% z*=1.645 α= .05 CI = 95% z*=1.96 α= .02 CI = 98%z*=2.326 α= .01 CI = 99% z*=2.576 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 9 Critical Values Again (cont.) Here are the traditional critical values from the Normal model: 1-sided 2-sided 0.05 1.645 1.96 0.01 2.28 2.575 0.001 3.09 3.29 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 10 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%. Test an appropriate hypothesis and state your conclusion. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 11 Comparing Proportions – Two Sample Test Women believe in ghosts more than men People who have been to jail believe in ghosts more than people who haven’t been to jail Women smoke more than men Women use facebook in the bathroom more than men Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 12 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 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 13 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 where SE pˆ1 pˆ 2 SE pˆ1 pˆ 2 pˆ1qˆ1 pˆ 2 qˆ2 n1 n2 The critical value z* depends on the particular confidence level, C, that you specify. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 14 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. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 15 Calculating the Pooled Proportion The pooled proportion is pˆ pooled Success1 Success2 n1 n2 where 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.) Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 16 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. Construct a 98% confidence interval for the difference between the two student groups. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 17 Inferences about Means – One Sample Test All Priuses have fuel economy > 50 mpg Ford Focuses get 5 mpg on average The average starting salary for ISU graduates >$100,000 The average cholesterol level for a person with diabetes is 240. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 18 Means: One-sample - Hypothesis One-sided alternatives Ha: μ>hypothesized value Ha: μ <hypothesized value Two-sided alternatives Ha: μ ≠ hypothesized value Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 19 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 n 1 where the standard error of the mean is y t SE y s SE y n The critical value tn*1 depends 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. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 20 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. We estimate the standard error with SE y s n Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 21 Means: One-sample – Sample Standard Deviation The standard deviation, s, is just the square root of the variance and is measured in the same units as the original data. y y 2 s n 1 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 22 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. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 23 Comparing Means – Two Sample Test The MPG for the Prius is greater than the MPG for the Ford Focus ISU male graduates have a greater starting salary than women The cholesterol levels are the same for people with and without diabetes. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 24 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 sample sizes and a special formula. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 25 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! Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 26 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 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 27 Means: Two-Sample – Standard Error Remember that, for independent random quantities, variances add. So, the standard deviation of the difference between two sample means is SD y1 y2 12 n1 22 n2 We still don’t know the true standard deviations of the two groups, so we need to estimate and use the standard error s12 s22 SE y1 y2 n1 n2 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 28 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. Interpret your confidence interval. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 29