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Chapter 13 Testing Hypotheses about Means Copyright ©2011 Brooks/Cole, Cengage Learning Hypothesis testing about: • a population mean • a population mean difference (paired data) • the difference between means of two populations Three Cautions: 1. Inference is only valid if the sample is representative of the population for the question of interest. 2. Hypotheses and conclusions apply to the larger population(s) represented by the sample(s). 3. If the distribution of a quantitative variable is highly skewed, consider analyzing the median rather than the mean – called nonparametric methods (Topic 2 on CD). Copyright ©2011 Brooks/Cole, Cengage Learning 2 13.1 Introduction to Hypothesis Tests for Means Steps in Any Hypothesis Test 1. Determine the null and alternative hypotheses. 2. Verify necessary data conditions, and if met, summarize the data into an appropriate test statistic. 3. Assuming the null hypothesis is true, find the p-value. 4. Decide whether or not the result is statistically significant based on the p-value. 5. Report the conclusion in the context of the situation. Copyright ©2011 Brooks/Cole, Cengage Learning 3 13.2 HT Module 3: Testing Hypotheses about One Mean Step 1: Determine null and alternative hypotheses 1. H0: m = m0 versus Ha: m m0 (two-sided) 2. H0: m = m0 versus Ha: m < m0 (one-sided) 3. H0: m = m0 versus Ha: m > m0 (one-sided) Remember a p-value is computed assuming H0 is true, and m0 is the value used for that computation. Copyright ©2011 Brooks/Cole, Cengage Learning 4 Step 2: Verify Necessary Data Conditions … Situation 1: Population of measurements of interest is approximately normal, and a random sample of any size is measured. In practice, use method if shape is not notably skewed or no extreme outliers. Situation 2: Population of measurements of interest is not approximately normal, but a large random sample (n 30) is measured. If extreme outliers or extreme skewness, better to have a larger sample. Copyright ©2011 Brooks/Cole, Cengage Learning 5 Continuing Step 2: The Test Statistic The t-statistic is a standardized score for measuring the difference between the sample mean and the null hypothesis value of the population mean: sample mean null value x m 0 t s standard error n This t-statistic has (approx) a t-distribution with df = n - 1. Copyright ©2011 Brooks/Cole, Cengage Learning 6 Step 3: Assuming H0 true, Find the p-value • For Ha less than, the p-value is the area below t, even if t is positive. • For Ha greater than, the p-value is the area above t, even if t is negative. • For Ha two-sided, p-value is 2 area above |t|. Copyright ©2011 Brooks/Cole, Cengage Learning 7 Steps 4 and 5: Decide Whether or Not the Result is Statistically Significant based on the p-value and Report the Conclusion in the Context of the Situation These two steps remain the same for all of the hypothesis tests considered in this book. Choose a level of significance a, and reject H0 if the p-value is less than (or equal to) a. Otherwise, conclude that there is not enough evidence to support the alternative hypothesis. Copyright ©2011 Brooks/Cole, Cengage Learning 8 Example 13.1 Normal Body Temperature What is normal body temperature? Is it actually less than 98.6 degrees Fahrenheit (on average)? Step 1: State the null and alternative hypotheses H0: m = 98.6 Ha: m < 98.6 where m = mean body temperature in human population. Copyright ©2011 Brooks/Cole, Cengage Learning 9 Example 13.1 Normal Body Temp (cont) Data: random sample of n = 16 normal body temps 98.4, 98.6, 98.8, 98.8, 98.0, 97.9, 98.5, 97.6, 98.4, 98.3, 98.9, 98.1, 97.3, 97.8, 98.4, 97.4 Step 2: Verify data conditions … Boxplot shows no outliers nor strong skewness. Sample mean of 98.2 is close to sample median of 98.35. Copyright ©2011 Brooks/Cole, Cengage Learning 10 Example 13.1 Normal Body Temp (cont) Step 2: … Summarizing data with a test statistic Key elements: Sample statistic: x = 98.200 (under “Mean”) s 0.497 Standard error: s.e.x 0.124 (under “SE Mean”) n 16 x m0 98.2 98.6 t 3.22 s 0.124 n Copyright ©2011 Brooks/Cole, Cengage Learning (under “T”) 11 Example 13.1 Normal Body Temp (cont) Step 3: Find the p-value From output: p-value = 0.003 From Table A.3: p-value is less than 0.004. Copyright ©2011 Brooks/Cole, Cengage Learning 12 Example 13.1 Normal Body Temp (cont) Step 4: Decide whether or not the result is statistically significant based on the p-value Using a = 0.05 as the level of significance criterion, the results are statistically significant because 0.003, the p-value of the test, is less than 0.05. In other words, we can reject the null hypothesis. Step 5: Report the Conclusion We can conclude, based on these data, that the mean temperature in the human population is actually less than 98.6 degrees. Copyright ©2011 Brooks/Cole, Cengage Learning 13 Rejection Region Approach Replaces Steps 3 and 4 with: Substitute Step 3: Find the critical value and rejection region for the test. Substitute Step 4: If the test statistic is in the rejection region, conclude that the result is statistically significant and reject the null hypothesis. Otherwise, do not reject the null hypothesis. Note: Rejection region method and p-value method will always arrive at the same conclusion about statistical significance. Copyright ©2011 Brooks/Cole, Cengage Learning 14 Rejection Region Approach Summary (use row of Table A.2 corresponding to df) For Example 13.1 Normal Body Temperature? Alternative was one-sided to the left, df = 15, and a = 0.05. Critical value from table A.2 is –1.75. Rejection region is t – 1.75. The test statistic was –3.22 so the null hypothesis is rejected. Same conclusion is reached. Copyright ©2011 Brooks/Cole, Cengage Learning 15 13.3 HT Module 4: Testing Hypotheses about Mean of Paired Differences Data: two variables for n individuals or pairs; use the difference d = x1 – x2. Parameter: md = population mean of differences Sample estimate: d = sample mean of the differences Standard deviation and standard error: sd = standard deviation of the sample of differences; sd s.e.d n Often of interest: Is the mean difference in the population different from 0? Copyright ©2011 Brooks/Cole, Cengage Learning 16 Steps for a Paired t-Test Step 1: Determine null and alternative hypotheses H0: md = 0 versus Ha: md 0 or Ha: md < 0 or Ha: md > 0 Watch how differences are defined for selecting the Ha. Step 2: Verify data conditions and compute test statistic Conditions apply to the differences. sample mean null value d 0 The t-test statistic is: t sd standard error n Steps 3, 4 and 5: Similar to t-test for a single mean. The df = n – 1, where n is the number of differences. Copyright ©2011 Brooks/Cole, Cengage Learning 17 Example 13.2 Effect of Alcohol Study: n = 10 pilots perform simulation first under sober conditions and then after drinking alcohol. Response: Amount of useful performance time. (longer time is better) Question: Does useful performance time decrease with alcohol use? Step 1: State the null and alternative hypotheses H0: md = 0 versus Ha: md > 0 where md = population mean difference between alcohol and no alcohol measurements if all pilots took these tests. Copyright ©2011 Brooks/Cole, Cengage Learning 18 Example 13.2 Effect of Alcohol (cont) Data: random sample of n = 10 time differences Step 2: Verify data conditions … Boxplot shows no outliers nor extreme skewness. Copyright ©2011 Brooks/Cole, Cengage Learning 19 Example 13.2 Effect of Alcohol (cont) Step 2: … Summarizing data with a test statistic Test of mu = 0.0 vs mu > 0.0 Variable N Mean StDev Diff 10 195.6 230.5 SE Mean T 72.9 2.68 P 0.013 Key elements: Sample statistic: d = 195.6 (under “Mean”) sd 230.5 Standard error: s.e.d 72.9 (under “SE Mean”) n 10 d 0 195.6 0 t 2.68 sd 72.9 n Copyright ©2011 Brooks/Cole, Cengage Learning (under “T”) 20 Example 13.2 Effect of Alcohol (cont) Step 3: Find the p-value From output: p-value = 0.013 From Table A.3: p-value is between 0.007 and 0.015. The value t = 2.68 is between column headings 2.58 and 3.00 in the table, and for df =9, the one-sided p-values are 0.015 and 0.007. Copyright ©2011 Brooks/Cole, Cengage Learning 21 Example 13.2 Effect of Alcohol (cont) Steps 4 and 5: Decide whether or not the result is statistically significant based on the p-value and Report the Conclusion Using a = 0.05 as the level of significance criterion, we can reject the null hypothesis since the p-value of 0.013 is less than 0.05. Even with a small experiment, it appears that alcohol has a statistically significant effect and decreases performance time. Copyright ©2011 Brooks/Cole, Cengage Learning 22 13.4 HT Module 5: Testing Hypotheses about Difference between Two Means Lesson 1: the General (Unpooled) Case Step 1: Determine null and alternative hypotheses H0: m1 – m2 = 0 versus Ha: m1 – m2 0 or Ha: m1 – m2 < 0 or Ha: m1 – m2 > 0 Watch how Population 1 and 2 are defined. Copyright ©2011 Brooks/Cole, Cengage Learning 23 Step 2: Verify data conditions and compute the test statistic. Both n’s are large or no extreme outliers or skewness in either sample. Samples are independent. The t-test statistic is: sample mean null value x1 x2 0 t standard error s12 s22 n1 n2 Steps 3, 4 and 5: Similar to t-test for one mean. Copyright ©2011 Brooks/Cole, Cengage Learning 24 Example 13.4 Effect of Stare on Driving Randomized experiment: Researchers either stared or did not stare at drivers stopped at a campus stop sign; Timed how long (sec) it took driver to proceed from sign to a mark on other side of the intersection. Question: Does stare speed up crossing times? Step 1: State the null and alternative hypotheses H0: m1 – m2 = 0 versus Ha: m1 – m2 > 0 where 1 = no-stare population and 2 = stare population. Copyright ©2011 Brooks/Cole, Cengage Learning 25 Example 13.3 Effect of Stare (cont) Data: n1 = 14 no stare and n2 = 13 stare responses Step 2: Verify data conditions … No outliers nor extreme skewness for either group. Copyright ©2011 Brooks/Cole, Cengage Learning 26 Example 13.3 Effect of Stare (cont) Step 2: … Summarizing data with a test statistic Sample statistic: x1 x2 = 6.63 – 5.59 = 1.04 seconds Standard error: s.e.( x1 x2 ) t x1 x2 0 1.04 0 2.41 s12 s22 n1 n2 s12 s22 1.36 2 0.822 2 0.43 n1 n2 14 13 0.43 Copyright ©2011 Brooks/Cole, Cengage Learning 27 Example 13.3 Effect of Stare (cont) Steps 3, 4 and 5: Determine the p-value and make a conclusion in context. The p-value = 0.013, so we reject the null hypothesis, the results are “statistically significant”. The p-value is determined using a t-distribution with df = 21 (df using Welch approximation formula) and finding area to right of t = 2.41. Table A.3 p-value is between 0.009 and 0.015. We can conclude that if all drivers were stared at, the mean crossing times at an intersection would be faster than under normal conditions. Copyright ©2011 Brooks/Cole, Cengage Learning 28 Lesson 2: Pooled Two-Sample t-Test Based on assumption that the two populations have equal population standard deviations: 1 2 Pooled standard deviation s p Pooled s.e.( x1 x2 ) s p n1 1s12 n2 1s22 n1 n2 2 1 1 n1 n2 sample mean null value x1 x2 0 t pooled standard error 1 1 s 2p n1 n2 Note: Pooled df = (n1 – 1) + (n2 – 1) = (n1 + n2 – 2). Copyright ©2011 Brooks/Cole, Cengage Learning 29 Example 13.7 Male and Female Sleep Times Q: Is there a difference between how long female and male students slept the previous night? Data: The 83 female and 65 male responses from students in an intro stat class. The null and alternative hypotheses are: H0: m1 – m2 = 0 versus Ha: m1 – m2 0 where 1 = female population and 2 = male population. Note: Sample sizes similar, sample standard deviations similar. Use of pooled procedure is warranted. Copyright ©2011 Brooks/Cole, Cengage Learning 30 Example 13.5 Male and Female Sleep Times Two-sample T for sleep [without “Assume Equal Variance” option] Sex Female Male N 83 65 Mean StDev SE Mean 7.02 1.75 0.19 6.55 1.68 0.21 95% CI for mu(f) – mu(m): (-0.10, 1.02) T-Test mu (f) = mu(m) (vs not =): T-Value = 1.62 P = 0.11 DF = 140 Two-sample T for sleep [with “Assume Equal Variance” option] Sex Female Male N 83 65 Mean 7.02 6.55 StDev 1.75 1.68 SE Mean 0.19 0.21 95% CI for mu(f) – mu(m): (-0.10, 1.03) T-Test mu (f) = mu(m) (vs not =): T-Value = 1.62 P = 0.11 DF = 146 Both use Pooled StDev = 1.72 Copyright ©2011 Brooks/Cole, Cengage Learning 31 13.5 Relationship Between Tests and Confidence Intervals For two-sided tests (for one or two means): H0: parameter = null value and Ha: parameter null value • If the null value is covered by a (1 – a)100% confidence interval, the null hypothesis is not rejected and the test is not statistically significant at level a. • If the null value is not covered by a (1 – a)100% confidence interval, the null hypothesis is rejected and the test is statistically significant at level a. Note: 95% confidence interval 5% significance level 99% confidence interval 1% significance level Copyright ©2011 Brooks/Cole, Cengage Learning 32 Example 13.9 Ear Infections and Xylitol 95% CI for p1 – p2 is 0.020 to 0.226 Reject H0: p1 – p2 = 0 and accept Ha: p1 – p2 > 0 with a = 0.025, because the entire confidence interval falls above the null value of 0. Note that the p-value for the test was 0.01, which is less than 0.025. Copyright ©2011 Brooks/Cole, Cengage Learning 33 13.6 Choosing an Appropriate Inference Procedure • Confidence Interval or Hypothesis Test? Is main purpose to estimate the numerical value of a parameter? … or to make a “maybe not/maybe yes” conclusion about a specific hypothesized value for a parameter? Copyright ©2011 Brooks/Cole, Cengage Learning 34 13.6 Choosing an Appropriate Inference Procedure • Determining the Appropriate Parameter Is response variable categorical or quantitative? Is there one sample or two? If two, independent or paired? Copyright ©2011 Brooks/Cole, Cengage Learning 35 13.8 Evaluating Significance in Research Reports 1. Is the p-value reported? If know p-value, can make own decision, based on severity of Type 1 error and p-value. 2. If word significant is used, determine whether used in everyday sense or in statistical sense only. Statistically significant just means that a null hypothesis has been rejected, no guarantee the result has real-world importance. 3. If you read “no difference” or “no relationship” has been found, determine whether sample size was small. Test may have had very low power because not enough data were collected to be able to make a firm conclusion. Copyright ©2011 Brooks/Cole, Cengage Learning 36 13.8 Evaluating Significance in Research Reports 4. Think carefully about conclusions based on extremely large samples. If very large sample size, even weak relationship or small difference can be statistically significant. 5. If possible, determine what confidence interval should accompany a hypothesis test. Intervals provide information about magnitude of effect as well as information about margin of error in sample estimate. 6. Determine how many hypothesis tests were conducted in study. Sometimes researchers perform multitude of tests, but only few achieve statistical significance. If all null hypotheses true, then ~1 in 20 tests will achieve statistical significance just by chance at the .05 level of significance. Copyright ©2011 Brooks/Cole, Cengage Learning 37