Chapter 5 Hypothesis Tests With Means of Samples Part 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. Accuracy of such a point estimate of the pop mean – ok, but not great –Our sample may be unrepresentative, etc.

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 dev =.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) and another 34% b/w 5.9 & 5.7 (-1 SD) –68% between 5.7 and 6.1  consider this a 68% confidence interval What does that mean?

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 (which is 2-tailed), 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 –Change these to raw scores for our example Using x = z(σ M ) + M, what scores do you get? What is the final 95% confidence interval for this example and what does it mean?

99% Confidence Intervals (analogous to alpha =.01) For 99% interval (2-tailed), 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 %. Change these to raw scores for our example –x = z(σ M ) + M, what do you get? –What is the 99% confidence interval here? What does it mean?

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 the mean from your null hyp (which is μ),  Reject Null. So what conclusions would we make in our example for the 95% CI? 99% CI?

1 vs. 2 tailed estimates Also note that we can calculate CIs for 1-tailed tests: –You will still calculate 2 scores to give a range of confidence. –What changes is the relevant z score… –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… Resulting 1-tailed 95% CI is… Narrower or wider interval than 2-tailed 95% CI?

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