1. Chapter 7 Hypothesis Tests With Means of Samples – Part 2 Oct. 6

2. Estimating μ  In estimating a population mean we have 2 options: –1) Point estimates give specific # –In the example from Tues (problem #9), we found a sample M=5.9 for the group of students who read about the accident w/ the word “crashed”. –Our estimate for μ (pop mean for students who get ‘crash’ wording) should = 5.9. It’s based on our previous sample, which is the best guess  Accuracy of such a point estimate of the pop mean – ok, but not great –Our sample may be unrepresentative, etc.

3. Estimating μ (cont.)  2) Interval estimates – provide range where you think pop mean may fall –Ex) Given M=5.9 (which = μ M), and std error =.2 (which is σ M), and assuming a normal curve… –We’d expect 34% of pop means to fall b/w 5.9 & 6.1 (+1 SD/Std error) and another 34% b/w 5.9 & 5.7 (-1 SD/Std error) –68% between 5.7 and 6.1  consider this a 68% confidence interval You can be 68% confident that a new population of students who get “crashed” manipulation would have a mean between 5.7 & 6.1

4. 95% Confidence Intervals (analogous to alpha =.05)  But 68% confident not that great…more interest in 95% or 99% confidence.  Standard to use 95% or 99% –For 95% interval, we’re left with 47.5% of scores (95/2) on each side of the mean up to our cutoff –Use normal curve table, find z=1.96 and –1.96 for those %s (note – this is 2-tailed!) –Change these to raw scores for our example (x = z(σ M ) + M), get: x = 1.96(.2) + 5.9 = 6.29… and x = -1.96(.2) + 5.9 = 5.51 –95% confident true pop mean for ‘crash’ pop lies b/w 5.51 and 6.29

5. 99% Confidence Intervals (analogous to alpha =.01)  For 99% interval, area of curve on each side of mean up to cutoff points = 49.5% (99/2), use normal table, find z=2.57 and –2.57 for those %. (note – this is 2-tailed!)  Change these to raw scores for our example (x = z(σ M ) + M), get: – x = 2.57(.2) + 5.9 = 6.41… and – x = -2.57(.2) + 5.9 = 5.39  99% confident true pop mean for ‘crash’ pop lies b/w 5.39 and 6.41

6. Confidence Intervals (CI)  Notice the wider interval for 99% compared to narrower interval for 95% –Wider  more likely you’re right and you include the actual mean in that interval  Can be used for hyp testing: –Here’s the Rule: if the CI does not contain mean from null hyp (which is μ),  Reject Null.

7. CI and Hypothesis Testing (cont.)  Our ex) – 95% conf int (5.51, 6.29) does not include 5.50 (see Tues example), so reject Null & conclude our 5.9 sample mean is unlikely to come from that population (it differs from the pop)  But 99% conf int does include 5.50, so sample group would not differ if we use.01 signif level

8. 1 vs. 2 tailed estimates  Also note that we can calculate CIs for 1-tailed tests: –95% CI for ‘crashed’ example: Z score cutoff will be 1.64, so use conversion formula: X = Z(σ M ) + M and then x = Z(σ M ) - M, so… –X = 1.64 (.2) + M = 6.23 and –X = -1.64 (.2) + M = 5.57 We are 95% confident for this 1 tailed test that the true pop mean is betw 5.57 and 6.23 Notice that a 95% 1-tailed test gives a narrower CI than the 95% 2-tailed test

9. Cutoff Scores for CI’s  As a shortcut, you may memorize or refer to these cutoff scores when computing CI’s – these will never change!  Cutoff scores most often used: –For a 95% CI, 1-tailed = 1.64 or –1.64 –95% CI, 2-tailed = 1.96 and –1.96 –99% CI, 1-tailed = 2.33 or –2.33 –99% CI, 2-tailed = 2.57 and –2.57

10. Activity 4 (cont.)  Finish #8 at end of Ch 7 from Tues.  In attempt to decrease reaction time, 25 women participate in training. After training, women have M=1.5 sec; gen population μ = 1.8 with σ =.5 1.(If you completed Tues. activity, you’ve already done this) Carry out 5 hyp testing steps (use alpha=.05) (and be sure to draw the comparison distribution to help you!) 2.(New!) Figure the 95% CI and interpret its meaning. Does it lead to same conclusion about rejecting/failing to reject the null as Part 1? Turn in this along with part 1 from Tues.