Guide to Chapter 11

This is your guide to what’s important in the chapter, with comments on some things that the chapter leaves out or doesn’t explain well. Page numbers refer to Sullivan, Michael, Fundamentals of Statistics 3/e (Pearson Prentice Hall, 2011), which is equivalent to the “second custom edition” for TC3.

Always check Corrections to Sullivan’s Fundamentals of Statistics, 3rd Edition for known mistakes in the textbook. Caution: your calculator is more accurate than the book because the book rounds values and uses tables. When the book disagrees with your answer, check the errata.

Big picture:  Previously we asked whether the difference between a sample statistic and some pre-identified number was too large for random chance. Now we ask whether the difference between two samples is too large for random chance (is statistically significant). You’ll also learn two-population confidence intervals.

MATH200A:  Use MATH200A part 2 to check for outliers, MATH200A part 5 to test for normality, and MATH200A part 6 to compute sample size.

Announce:  Fall and spring students, you should have your project topic already set. Make sure you discuss your Field Project Plan with me before next week’s deadline.

11.1  Inference about Dependent Samples

Testing new corn versus standard corn for yield. Can you see a problem with the sample in Western New York that’s not a problem with the sample in Central New York?
source: Dabes & Janik’s Statistics Manual (1999)

509–10 Understand the difference between independent samples (unpaired data) and dependent samples (paired data). Which way do you want to design an experiment, when possible? Why?

Randomization (page 48) produces independent samples (unpaired data).

Pairs can even be the same individual. Examples: heart rate before and after drinking coffee; mood before and after taking an antidepressant.

510The terminology in the box is wrong, though the requirements are right. You have a mean difference μ_d. The difference of means μ₁−μ₂ comes in section 11.2.

510–1 These are the same six steps (plus RC) that you know and love. The only differences are (a) in step 1 you define d and write hypotheses on μ_d; (b) in RC and steps 3–4 you work with the d’s, not the original data.

To compute the d’s, type the subtraction directly into each cell of your statistics list. Or, if you like calculator tricks, use the optional technique in List Operations for Paired t Test on TI-83/84.

511–4 Work Example 2 with TI-83. Remember, the solution is written out for you.

512 Understand the connection between which way you subtract and whether H₁ contains < or >. This is why you must state up front which way you will subtract, by defining d in step 1.

516 Example 4: work CI with TI-83

Caution: You’re estimating a difference, so phrase your interpretation that way: give size and direction of the difference. More examples: Confidence Intervals for Two Populations

Comment on the last paragraph of the example: a CI is the flip side of a two-tailed test. So this 95% CI says that the difference is statistically significant in a two-tailed test at α = 0.05. Example 2 that we did was a one-tailed test at α = 0.05, which would correspond to a two-tailed test at α = 0.10, so this particular CI actually provides a stricter test. See Hypothesis Test by Confidence Interval, which was also in the Chapter 10 handouts.

516 practice problems 1–4

11.2  Inference about Independent Samples

522 Know the requirements.

Sample sizes need not be equal but should not be very different. (Rupert G. Miller., Jr., Beyond ANOVA: Basics of Applied Statistics [Wiley, 1986], page 57, says that if sample sizes are equal or nearly equal, the two-sample t test can tolerate one standard deviation being as large as double the other, so “it pays to balance the experiment as closely as possible.”)

optional: Look at the t statistic and compare to 1-sample t. See the formula for standard error? It is what it is because variances add when you add or subtract population data. (It’s interesting, but we don’t actually use it in computations.)

523 Again, the same old six steps plus RC. However, (a) you must always identify populations 1 and 2 (which your book doesn’t always do), and (b) in steps 3–4 you use a 2-SampTTest with Pooled:No.

523–6 Work Example 1 using the TI-83. Remember, the solution is written out for you.

525 Don’t worry about the formula for degrees of freedom at the bottom; in two-sample t tests everyone always lets the computer or calculator compute df.

528 Use TI-83 for Example 3 (CI, same data).

Caution: You’re estimating a difference, so phrase your interpretation that way: give size and direction of the difference. More examples: Confidence Intervals for Two Populations

528–9 You can skip the paragraphs about pooling. We always use Pooled:No.

11.3  Inference about 2 Pop. Proportions

535 Know the requirements, in particular the use of pooled p̂, which the 2-PropZTest computes for you. (Note: samples must be independent. Sample sizes need not be equal but should not be very different.)

The z statistic is just an extension of Case 2, but you’ll never need to calculate it by hand.

536 Once again, the same six steps plus RC. You must identify populations 1 and 2 as part of step 1.

536–8 We won’t do Example 1 in class; do it on your own with your calculator. Instead, we’ll use the doctors’ health study on aspirin and heart attacks.

539–40 Do Example 3 on your own; in class we’ll do a CI on the doctors’ health study.

For this case, you use blended p̂ to test requirements for HT but individual p̂₁ and p̂₂ to test requirements for CI. However, if you’ve just done a HT and the requirements are met, you don’t need to re-test requirements when computing a CI from the same samples.

Caution: You’re estimating a difference, so phrase your interpretation that way: give size and direction of the difference. More examples: Confidence Intervals for Two Populations

Look at Case Study: Gardasil Vaccine on your own.

540–2 Skip objective 3.

543–4 Use MATH200A part 6 to find needed sample size. As before, use prior estimates if you have them; otherwise use .5 for both.

Practice with Example 5.

What’s New

• 15 Jan 2012: Add reference to Miller’s statement about unequal s.d.
• 20 Apr 2011: Four or five minor edits for clarity, and a cross reference to Hypothesis Test by Confidence Interval
• 25 Jan 2011: Start over for the third edition of the textbook.
• (intervening changes suppressed)
• 24 Apr 2008: new document (MS-Word format)