8.2 – Estimating a Population Proportion P, Constructing Confidence Intervals for Proportions

8.2 – Estimating a population proportion p Constructing confidence intervals for proportions Assumptions • The sample is a simple random sample • The normal distribution can be used to approximate the distribution of sample proportions because np ≥ 15 and n(1-p) ≥ 15 are both satisfied. Note: Other books use np ≥ 10 and n(1-p)≥10 and this will be the criteria used in many of my solution documents. Procedure for Constructing a Confidence Interval for p 1) Verify that the assumptions are satisfied. 2) Evaluate the margin of error E. E = zc 3) Construct the interval: p•q n p−E p p+E; Round-Off Rule for Confidence Interval Estimates of p: ( p − E, p + E ) 3 significant digits Using the TI-83 to Construct Confidence Intervals for p: STAT>>TESTS use A:1-propZInt. Notice: x must be a whole number; if you have to calculate it, round it to the nearest whole x number. Since p = , then x = p n n 8.4 - SAMPLE SIZE FORMULAS for estimating population proportions • We have a p-hat available from a prior study p (1 − p ) z 2 n= where E is the margin of error E2 (Note, our book uses the symbol m for the margin of error) • No prior p-hat is available; we use p-hat = 0.5 and the formula becomes 0.25 z 2 n= E2 1 1) In an October 2003 poll conducted by the Gallup Organization, 684 of 1006 randomly selected adults aged 18 years old or older stated they think the government should make partial birth abortions illegal, except in cases necessary to save the life of the mother. a) Obtain a point estimate for the proportion of adults aged 18 or older who think the government should make partial birth abortions illegal, except in cases necessary to save the life of the mother. b) Verify that the requirements for constructing a confidence interval about p-hat are satisfied. c) Construct a 98% confidence interval for the proportion of adults aged 18 or older who think the government should make partial birth abortions illegal, except in cases necessary to save the life of the mother. Interpret the interval. d) We are _____% confident that the true proportion of adults aged 18 or older who think the government should make partial birth abortions illegal, except in cases necessary to save the life of the mother is between _________% and __________% e) With ______% confidence we can say that ________% adults aged 18 or older think the government should make partial birth abortions illegal, except in cases necessary to save the life of the mother with a margin of error of ________% f) The statement “98% confident” means that, if 100 samples of size _____ were taken, about _____ of the intervals will contain the parameter p and about ____ will not. g) For ______% of intervals constructed with this method, the sample percentage would not differ from the actual population proportion by more than ______ g) How many more adults should be included in the sample to be 98% confident that a point estimate p-hat will be within 1% of p? Use a p-hat of 0.68. h) If no preliminary sample is taken to estimate p, how large a sample is necessary to be 98% confident that the point estimate p-hat will be within a distance of 0.01 from p? 2 2) In a Gallup poll, 491randomly selected adults were asked whether they are in favor of the death penalty for a person convicted of murder, and 65% of them said that they were in favor. Construct a 90% confidence interval estimate for the proportion of adults who are in favor of the death penalty. a) What is the point estimate for p? b) Verify that the requirements for constructing a confidence interval about p-hat are satisfied. c) Construct a 90% confidence interval for the proportion of adults who are in favor of the death penalty. d) We are _____% confident that the ____________________________________________ is between _________ and __________ e) With ______% confidence we can say that ________% of adults __________________ with a margin of error of ________% f) The statement “90% confident” means that, if 100 samples of size _____ were taken, about _____ of the intervals will contain the parameter p and about ____ will not. g) For ______% of intervals constructed with this method, the sample proportion would not differ from the actual population proportion by more than ______ g) How many more adults should be included in the sample to be 90% confident that a point estimate p-hat will be within 3% of p? Use a p-hat of 0.65. h) If no preliminary sample is taken to estimate p, how large a sample is necessary to be 90% confident that the point estimate p-hat will be within a distance of 0.03 from p? 3