Copyright ©2011 Brooks/Cole, Cengage Learning Testing Hypotheses about Difference Between Two Means

Copyright ©2011 Brooks/Cole, Cengage Learning 2 Testing Hypotheses about Difference between Two Means Step 1: Determine null and alternative hypotheses H 0 :  1 –  2 =  versus H a :  1 –  2   or H a :  1 –  2  Watch how Population 1 and 2 are defined. Lesson 1: the General (Unpooled) Case

Copyright ©2011 Brooks/Cole, Cengage Learning 3 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: Steps 3, 4 and 5: Similar to t-test for one mean.

Examples From Final Exam Review Sheet III Copyright ©2011 Brooks/Cole, Cengage Learning 4

5 Example: Effect of Stare on Driving Question: Does stare speed up crossing times? Step 1: State the null and alternative hypotheses H 0 :  1 –  2 =  versus H a :  1 –  2 >  where 1 = no-stare population and 2 = stare population. 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.

Copyright ©2011 Brooks/Cole, Cengage Learning 6 Example: Effect of Stare (cont) Data: n 1 = 14 no stare and n 2 = 13 stare responses Step 2: Verify data conditions … No outliers nor extreme skewness for either group.

Copyright ©2011 Brooks/Cole, Cengage Learning 7 Example: Effect of Stare (cont) Step 2: … Summarizing data with a test statistic Sample statistic: = 6.63 – 5.59 = 1.04 seconds Standard error:

Copyright ©2011 Brooks/Cole, Cengage Learning 8 Example: 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 = Table A.3  p-value is between and 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 9 Lesson 2: Pooled Two-Sample t-Test Based on assumption that the two populations have equal population standard deviations: Note: Pooled df = (n 1 – 1) + (n 2 – 1) = (n 1 + n 2 – 2).

Copyright ©2011 Brooks/Cole, Cengage Learning 10 Guidelines for Using Pooled t-Test If sample sizes are equal, pooled and unpooled standard errors are equal and so t-statistic is same. If sample standard deviations are similar, assumption of common population variance is reasonable and pooled procedure can be used. If sample sizes are very different, pooled test can be quite misleading unless sample standard deviations similar. If sample sizes very different and smaller standard deviation accompanies larger sample size, do not recommend using pooled procedure. If sample sizes are very different, standard deviations are similar, and larger sample size produced the larger standard deviation, pooled t-test is acceptable and will be conservative.

Copyright ©2011 Brooks/Cole, Cengage Learning 11 The null and alternative hypotheses are: H 0 :  1 –  2 =  versus H a :  1 –  2   where 1 = female population and 2 = male population. Example: Male and Female Sleep Times Data:The 83 female and 65 male responses from students in an intro stat class. Note: Sample sizes similar, sample standard deviations similar. Use of pooled procedure is warranted. Q: Is there a difference between how long female and male students slept the previous night?

Copyright ©2011 Brooks/Cole, Cengage Learning 12 Example: Male and Female Sleep Times Two-sample T for sleep [without “Assume Equal Variance” option] Sex N Mean StDev SE Mean Female Male % 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 N Mean StDev SE Mean Female Male % 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 13 Relationship Between Tests and Confidence Intervals For two-sided tests (for one or two means): H 0 : parameter = null value and H a : parameter  null value Note: 95% confidence interval  5% significance level 99% confidence interval  1% significance level If the null value is covered by a (1 –  )100% confidence interval, the null hypothesis is not rejected and the test is not statistically significant at level . If the null value is not covered by a (1 –  )100% confidence interval, the null hypothesis is rejected and the test is statistically significant at level .

Copyright ©2011 Brooks/Cole, Cengage Learning 14 Confidence Intervals and One-Sided Tests If the null value is covered by the interval, the test is not statistically significant at level . For the alternative H a : parameter > null value, the test is statistically significant at level  if the entire interval falls above the null value. For the alternative H a : parameter < null value, the test is statistically significant at level  if the entire interval falls below the null value. When testing the hypotheses: H 0 : parameter = null value versus a one-sided alternative, compare the null value to a (1 – 2  )100% confidence interval:

Copyright ©2011 Brooks/Cole, Cengage Learning 15 Example: Ear Infections and Xylitol 95% CI for p 1 – p 2 is to Reject H 0 : p 1 – p 2 =  and accept H a : p 1 – p 2 >  with  = 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

SUPPLEMENTARY NOTES The following slides are for your personal consumption. The concepts are not part of the material for the final examination. Copyright ©2011 Brooks/Cole, Cengage Learning 16

Copyright ©2011 Brooks/Cole, Cengage Learning 17 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 18 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 19 Effect Size Effect size is a measure of how much the truth differs from chance or from a control condition. Effect size for a single mean: Effect size for comparing two means:

Copyright ©2011 Brooks/Cole, Cengage Learning 20 Estimating Effect Size Estimated effect size for a single mean: Estimated effect size for comparing two means: Relationship: Test statistic = Size of effect  Size of study

Copyright ©2011 Brooks/Cole, Cengage Learning 21 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 22 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.