Guide to Chapter 10
Copyright © 2008–2012 by Stan Brown, Oak Road Systems
Copyright © 2008–2012 by Stan Brown, Oak Road Systems
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: Now we turn to the “meat” of inferential statistics: hypothesis tests a/k/a significance tests. These test a claim (null hypothesis) by asking, If the null is true, could chance account for the sample we got, or is the sample too unusual? If it’s too unusual, we declare the null false and the alternative true.
You might still find it helpful to go through the book, just for a different perspective on these admittedly difficult concepts. If you do, you can use the notes below to help you. I recommend you work through sections 10.2, 10.3, 10.1, and 10.4 in that order.
Refer to Inferential Statistics: Basic Cases while you’re learning the requirements and calculator procedures. (You can use it in addition to your own crib sheet on all remaining quizzes and the exam.)
MATH200A: Use MATH200A part 2 to check for outliers and MATH200A part 5 to test for normality.
Fall and spring students, start thinking about a topic for your Field Project. (Please read the assignment before asking questions.)
463 Historical note: who dreamed up this stuff?
463 Always verify requirements as part of your HT. If n < 30, check sample is ND with no outliers, using MATH200A part 5 and MATH200A part 2.
464 know the meaning of “statistically significant”
464–5
Reminder: skip classical approach
465We compute probabilities in a normal distribution using
normalcdf, not table lookup. However, you’ll
soon see that the calculator covers everything for you in
Z-Test.
466–9 skip objective 2
469 Take time to understand the p-value. (See also handout, What Does the p-Value Mean?)
469 μo in step 1 is the claimed value of the population mean, or the value you are testing against. It is not the sample mean. Never, never look at the data before writing your hypotheses.
470–73 Work Example 3 using calculator.
Know the seven steps of HT and number them: see the handout, Hypothesis Tests: Six Steps (Plus One).
Show your work in steps 3–4. Write down screen name and all inputs before hitting Calculate, then from the output screen write down everything that isn’t a duplicate of the input screen.
Round the test statistic (z or t) to 2 decimals, p-value to four.
Watch for a negative exponent on p-value! Convert to decimal and use p<0.0001 if appropriate.
471 p-value in step 4 is 0.0827.
Caution! Step 6 says, “There is not sufficient evidence ... that students who take at least four years of English score better on the SAT math reasoning exam.” While that is literally true, it is only half of the truth. A careless or unsophisticated reader can come away with the idea that four years of English is no help on the math SAT, but that has not been proved (and can’t be).
On “fail to reject H0”, always write a conclusion in neutral language. In this case, that could be “There is not sufficient evidence ... to determine whether students who take at least four years of English score better on the SAT math reasoning exam or not. See Proper Conclusions to Your Hypothesis Tests.
472 Read the “Caution” paragraphs carefully.
Note also the statement that in practice we put p-values in the step 6 conclusion statements. I think that makes sense, and I don’t know why your book doesn’t do it.
472–3 Work Example 4 with calculator. Remember to check requirements.
p-value is 0.0071.
Step 6 concludes, correctly, that cell-phone bills have changed. But when you have a statistically significant difference, as you do here, you can go further and say whether it is an increase or a decrease. You do that by comparing the sample mean to μo. Then you can add a sentence, “In fact, the average bill has increased.” Again, you can do this on a two-tailed test only when p < α and you reject H0. See One-Tailed or Two-Tailed Hypothesis Test?.
474 Understand the relation between a 2-tailed HT at significance level α and a CI at (1−α).
475 “Statistically significant” may not be significant in ordinary English, particularly when you have a large sample.
This is the same deal as with CI — use a T-Test when σ is unknown.
Caution! A lot of students just think about “the” standard deviation and end up using a z test where they should use a t test. Don’t make this rookie mistake. Never ask yourself, “do I know the standard deviation?” Always ask yourself, “do I know the standard deviation of the population?” If the answer is no, and usually it is, do a t test not a z test.
See Top 10 Mistakes of Hypothesis Tests.
488–9 Practice with #16 and #21.
456 A hypothesis is a claim or statement about a population — initially words, then symbols
456–7 H0 is always “nothing going on here”, H1 is “something is different”
(Ha sometimes used for H1)
H0 always contains =; H1 always contains <, >, or ≠
terms: left-, right, one-, two-tailed tests
457 Work Example 2 together
458 Know the meaning of Type I and II errors (the chart helps).
read the “In Other Words”
remember Type I: rejecting the null when it’s actually true
Type II is often a missed opportunity
459 Example 3 is good for Type I versus Type II error
459–60 We choose α, we don’t compute it.
Note: significance level, not confidence level.
Once α is chosen, β is beyond our control (other than larger sample size).
460 Writing conclusions: what the book says, but neutral language on “fail to reject H0”. See Proper Conclusions to Your Hypothesis Tests.
460–1 Practice “reject” and “fail to reject” conclusions on #18b and #20.
492 Know the requirements — they’re almost the same as for CI about a proportion. (In item 2, you use po not p̂, since you assume that H0 is true until it’s disproved. Item 3, independence, is really just 20n < N, same as for CI.)
493 The test statistic is z again.
494–5 Practice with Example 1 on your own, and compare to the Hypothesis Tests of a Proportion handout. Note! This is a left-tailed test, not a right-tailed.
495–6 Do Example 2 in class.
497–8 If requirements 1 and 3 are met but npo(1−po) is substantially below 10, you can compute the p-value using MATH200A part 3.
Example: Lansing sewer
In 2006–2008 there was controversy about creating a sewer district in south Lansing, where residents have had their own septic tanks for years. The Sewer Committee sent out an opinion poll to every household in the proposed sewer district. In a letter to the editor, published 3 Feb 2007 in the Ithaca Journal, John Schabowski wrote, in part:
The Jan. 4 Journal article about the sewer reported that “only” 380 of 1366 households receiving the survey responded, with 232 against it, 119 supporting it, and 29 neutral. ... The survey results are statistically valid and accurate for predicting that the sewer project would be voted down by a large margin in an actual referendum.
Can you do a hypothesis test to show that more than half of Lansing households in the proposed district were against the sewer project?
In teams of two, do the M&Ms Lab: Inferences for One Population.
This page is used in instruction at Tompkins Cortland Community College in Dryden, New York; it’s not an official statement of the College. Please visit www.tc3.edu/instruct/sbrown/ to report errors or ask to copy it.
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