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March 8, 2006, Chapter 18, Population proportions
Read Chapter 18.
Due March 10th: 16: 19, 26, 27, 28, 37. 17: 5, 6, 11, 14, 16, 23, 24, 28, plus excel assignment. 17.41-43.
Midterm issues Notice that on multiple choice questions, I am trying to get you to demonstrate knowledge about the meaning of what you are doing rather than just the steps.
• 1C, 1D.
- 1H: meaning of P -value!
- 2: sampling distribution (Chapter 10)
- 5a: identifying parameter you are testing. Usually a population mean, but need to know what population and what variable. (Sometimes a population proportion or a difference of means.)
- 6: population standard deviation σ vs. sample standard deviation s.
Review: Matched pairs vs. two-sample
Matched pairs when:
- can do two different treatments to the same sample or
- take pairs of individuals and randomly assign one individual from each pair to one treatment and other to other treatment
Two sample test when:
- comparing two samples from different populations or
- comparing two treatments which can’t both be applied to the same sample.
- We calculate our P -value. Since Ha is μ 1 − μ 2 < 0 , we wish to look for P (t ≤ − 1 .644). We use t(30) since 31 is our smaller sample size. (see p. 452 for a more accurate way to determine degrees of freedom).
From the calculator, P (t ≤ − 1 .644) =. 0553.
- We draw our conclusion: If we assume H 0 is true, then the probability of seeing samples like the ones we have is. 0553. This is moderately low, so our assumption that H 0 was true is probably wrong. So, this is moderate evidence that boys score higher on IQ tests than girls.
- We can ask the calculator to do the test for us. This is under STAT ,
TESTS, 4:2-SampTTest. We get df = 56. 93 , t = − 1. 64 , P =. 053.
- We can also ask Excel to do a two-sample t-test. Tools , Data Analysis... ,
then t-Test: Two sample Assuming Unequal Variances.
In either case, if we use software we still need to do step 4 (conclusion) above. And we need to do it carefully, because we’ve possibly lost track of what all our numbers mean.
When using two-sample t-tests
- if n 1 + n 2 < 15 , want data to be approximately normal (symmetric, no outliers, single peaked).
- 15 ≤ n 1 + n 2 ≤ 40 want data to have no strong outliers, not strongly skewed.
- 40 ≤ n 1 + n 2 even skewed data is OK.
Confidence Intervals (Plus Four)
In theory our estimates based on samples of size n should of the form p is p ˆ ± z∗
pˆ(1 − pˆ)n.
Unfortunately because of the error involved in how close this distribution is to normal, and the fact that we’re approximating our standard deviation, this isn’t very accurate. Instead use plus four method:
- Pick sample, size n ≥ 10.
p ˜ =
number of successes + 2 n + 4
- Find z∗^ as usual corresponding to desired confidence level.
- Confidence interval for p is
p ˜ ± z∗
p˜(1 − p˜) n + 4
Example (left-handed)
p is percent of young adults in US who are left handed. Estimate with 95% confidence based on class sample.
- Ask for 95% confidence, so z∗^ = 1. 96
- Sample = people in class. Take as “random” sample of population of young adults.
- p˜ = number left-handed number in class+4+
- Calculate p is p˜ ± 1. 96 ·
p˜(1 − p˜)/n with 95% confidence.
- So we reject H 0 and conclude that fewer than 49% of registered voters would vote for Bush in November 2005.
