Confidence Intervals for Proportions Sections 22.2, 22.3
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1 Confidence Intervals for Proportions Sections 22.2, 22.3 Lecture 41 Robb T. Koether Hampden-Sydney College Mon, Apr 4, 2016 Robb T. Koether (Hampden-Sydney College)Confidence Intervals for ProportionsSections 22.2, 22.3 Mon, Apr 4, / 18
2 Outline 1 Confidence Intervals for p 2 Determining the Sample Size 3 Assignment Robb T. Koether (Hampden-Sydney College)Confidence Intervals for ProportionsSections 22.2, 22.3 Mon, Apr 4, / 18
3 Outline 1 Confidence Intervals for p 2 Determining the Sample Size 3 Assignment Robb T. Koether (Hampden-Sydney College)Confidence Intervals for ProportionsSections 22.2, 22.3 Mon, Apr 4, / 18
4 Confidence Intervals for p The standard deviation of ˆp is not know p. p(1 p), but in practice we do n Robb T. Koether (Hampden-Sydney College)Confidence Intervals for ProportionsSections 22.2, 22.3 Mon, Apr 4, / 18
5 Confidence Intervals for p p(1 p) The standard deviation of ˆp is, but in practice we do n not know p. We use ˆp as an estimator of p. Robb T. Koether (Hampden-Sydney College)Confidence Intervals for ProportionsSections 22.2, 22.3 Mon, Apr 4, / 18
6 Confidence Intervals for p p(1 p) The standard deviation of ˆp is, but in practice we do n not know p. We use ˆp as an estimator of p. This gives us the standard error of ˆp: ˆp(1 ˆp) SEˆp =. n Robb T. Koether (Hampden-Sydney College)Confidence Intervals for ProportionsSections 22.2, 22.3 Mon, Apr 4, / 18
7 Confidence Intervals for p Confidence intervals for p follow the usual pattern: (point estimate) ± (margin of error). Robb T. Koether (Hampden-Sydney College)Confidence Intervals for ProportionsSections 22.2, 22.3 Mon, Apr 4, / 18
8 Confidence Intervals for p Confidence intervals for p follow the usual pattern: (point estimate) ± (margin of error). The margin of error is margin of error = z (standard error). Robb T. Koether (Hampden-Sydney College)Confidence Intervals for ProportionsSections 22.2, 22.3 Mon, Apr 4, / 18
9 Confidence Intervals for p Confidence intervals for p follow the usual pattern: (point estimate) ± (margin of error). The margin of error is margin of error = z (standard error). The standard error is ˆp(1 ˆp) standard error =. n Robb T. Koether (Hampden-Sydney College)Confidence Intervals for ProportionsSections 22.2, 22.3 Mon, Apr 4, / 18
10 Confidence Intervals for p Thus, the confidence interval is ˆp(1 ˆp) ˆp ± z. n Robb T. Koether (Hampden-Sydney College)Confidence Intervals for ProportionsSections 22.2, 22.3 Mon, Apr 4, / 18
11 Confidence Intervals for p The normality assumption (use of z ) is justified provided the sample contains At least 15 members with the property, and At least 15 members without the property. Robb T. Koether (Hampden-Sydney College)Confidence Intervals for ProportionsSections 22.2, 22.3 Mon, Apr 4, / 18
12 Example Example (Confidence Intervals for p) A recent Quinnipiac poll found that 43% of 834 Republicans surveyed support Trump. Robb T. Koether (Hampden-Sydney College)Confidence Intervals for ProportionsSections 22.2, 22.3 Mon, Apr 4, / 18
13 Example Example (Confidence Intervals for p) A recent Quinnipiac poll found that 43% of 834 Republicans surveyed support Trump. Find a 95% confidence interval for the true proportion who support Trump. Robb T. Koether (Hampden-Sydney College)Confidence Intervals for ProportionsSections 22.2, 22.3 Mon, Apr 4, / 18
14 Example Example (Confidence Intervals for p) We have n = 834 and ˆp = Robb T. Koether (Hampden-Sydney College)Confidence Intervals for ProportionsSections 22.2, 22.3 Mon, Apr 4, / 18
15 Example Example (Confidence Intervals for p) We have n = 834 and ˆp = The standard error is Robb T. Koether (Hampden-Sydney College)Confidence Intervals for ProportionsSections 22.2, 22.3 Mon, Apr 4, / 18
16 Example Example (Confidence Intervals for p) We have n = 834 and ˆp = The standard error is ˆp(1 ˆp) n = (0.43)(0.57) 843 Robb T. Koether (Hampden-Sydney College)Confidence Intervals for ProportionsSections 22.2, 22.3 Mon, Apr 4, / 18
17 Example Example (Confidence Intervals for p) We have n = 834 and ˆp = The standard error is ˆp(1 ˆp) n (0.43)(0.57) = 843 = Robb T. Koether (Hampden-Sydney College)Confidence Intervals for ProportionsSections 22.2, 22.3 Mon, Apr 4, / 18
18 Example Example (Confidence Intervals for p) We have n = 834 and ˆp = The standard error is ˆp(1 ˆp) n (0.43)(0.57) = 843 = = Robb T. Koether (Hampden-Sydney College)Confidence Intervals for ProportionsSections 22.2, 22.3 Mon, Apr 4, / 18
19 Example Example (Confidence Intervals for p) The confidence interval is ˆp ± z ˆp(1 ˆp) n = 0.43 ± (1.960)(0.017) = 0.43 ± Robb T. Koether (Hampden-Sydney College)Confidence Intervals for ProportionsSections 22.2, 22.3 Mon, Apr 4, / 18
20 Example Example (Confidence Intervals for p) Recompute the confidence for n = 100. What is the effect? Robb T. Koether (Hampden-Sydney College)Confidence Intervals for ProportionsSections 22.2, 22.3 Mon, Apr 4, / 18
21 Example Example (Confidence Intervals for p) Recompute the confidence for n = 100. What is the effect? Recompute the confidence for n = What is the effect? Robb T. Koether (Hampden-Sydney College)Confidence Intervals for ProportionsSections 22.2, 22.3 Mon, Apr 4, / 18
22 Example Example (Confidence Intervals for p) Recompute the confidence for n = 100. What is the effect? Recompute the confidence for n = What is the effect? Recompute the confidence using ˆp = What is the effect? Robb T. Koether (Hampden-Sydney College)Confidence Intervals for ProportionsSections 22.2, 22.3 Mon, Apr 4, / 18
23 Example Example (Confidence Intervals for p) Recompute the confidence for n = 100. What is the effect? Recompute the confidence for n = What is the effect? Recompute the confidence using ˆp = What is the effect? Recompute the confidence using ˆp = What is the effect? Robb T. Koether (Hampden-Sydney College)Confidence Intervals for ProportionsSections 22.2, 22.3 Mon, Apr 4, / 18
24 Outline 1 Confidence Intervals for p 2 Determining the Sample Size 3 Assignment Robb T. Koether (Hampden-Sydney College)Confidence Intervals for ProportionsSections 22.2, 22.3 Mon, Apr 4, / 18
25 Determining the Sample Size We just saw that the margin of error depends on The population proportion, and The level of confidence, and The sample size. How can we achieve a specific margin of error, say 1%? Robb T. Koether (Hampden-Sydney College)Confidence Intervals for ProportionsSections 22.2, 22.3 Mon, Apr 4, / 18
26 Determining the Sample Size We cannot change the population proportion. We do not want to lower the level of confidence. The only thing left is to increase the sample size. How large should it be? Robb T. Koether (Hampden-Sydney College)Confidence Intervals for ProportionsSections 22.2, 22.3 Mon, Apr 4, / 18
27 Determining the Sample Size The formula for the margin of error is ˆp(1 ˆp) margin of error = z. n Let m be the margin of error and solve the equation for n. We get: ( ) z 2 n = ˆp(1 ˆp). m Robb T. Koether (Hampden-Sydney College)Confidence Intervals for ProportionsSections 22.2, 22.3 Mon, Apr 4, / 18
28 Example Example Determining the Sample Size Suppose Donald Trump has the support of about 42% of registered Republicans. How large must a sample be in order for the margin of error of a 95% confidence interval to be 3%? For a margin of error of 1%? What if we did not assume that p = 0.42? Robb T. Koether (Hampden-Sydney College)Confidence Intervals for ProportionsSections 22.2, 22.3 Mon, Apr 4, / 18
29 Outline 1 Confidence Intervals for p 2 Determining the Sample Size 3 Assignment Robb T. Koether (Hampden-Sydney College)Confidence Intervals for ProportionsSections 22.2, 22.3 Mon, Apr 4, / 18
30 Assignment Assignment Read Section 22.1, Apply Your Knowledge: 1, 2, 3, 4. Check Your Skills: 15, 16, 17, 18. Exercises 26, 27, 30, 31, 33. Robb T. Koether (Hampden-Sydney College)Confidence Intervals for ProportionsSections 22.2, 22.3 Mon, Apr 4, / 18
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