Inference for a Population Proportion Section 121 AP Registration Deadline March 17 th Late Fee 50 March 18 th 24 th Financial Aid Application Due March 1 st Remember Conditions for Inference ID: 768350
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Inference for a Population Proportion Section 12.1 AP Registration Deadline: March 17 th Late Fee ($50): March 18 th – 24 th Financial Aid Application Due: March 1 st
Remember Conditions for Inference Data are an SRS from the population of interest. Observations are independent (pop. ≥ 10*n) Sampling Distribution is approx. normal Today, we’re dealing with proportions, so np ≥ 10 and n (1- p ) ≥ 10.
Standard Error Replace standard deviation by the standard error of (or standard deviation of ) To get a confidence interval of the form Estimate ± z* SE
Inference for a Population Proportion Draw an SRS of size n from a large population with unknown proportion p of successes. An approximate level C confidence interval for p is where z* is the upper (1 – C)/2 standard normal critical value.
Remember: State P lan D o Conclude Statistics Problems Demand Consistency!!!
Example 1 A Gallup Poll found that 28% of a SRS of 682 American adults expect to inherit money. Construct a 90% Confidence interval for the true proportion. State: know what parameters we’re estimating & at what confidence level We want to estimate p = the true proportion of US adults who expect to inherit $ with 90% confidence.
Example 1 Plan: choose method & check conditions Method: Proportions Conditions:Random:Independent:Normal:Assume Gallup used correct sampling proceduresn = 682, the population of adults is much larger than 6820 (pop. ≥ 10*n), so assume independence. sampling distribution of is approx. normal
Example 1 Do: if conditions are met, perform calculations .
Example 1 Conclude: interpret the interval in the context of the problem We are 90% confident that the true percentage is between 25.17% and 30.83%.
YOUR TURN!!! The New York Times and CBS News conducted a nationwide poll of 1048 randomly selected 13- to 17-year-olds. Of these teenagers, 692 had a television in their room. We will act as if the sample were an SRS. Construct a 95% confidence interval for the proportion of all people in this age group who have a TV in their room.
!!!! We are trying to estimate the population proportion of teenagers who have a TV in their room at a 95% confidence level. Method: proportions, Conditions: SRS: Yes! Independent: Population of teenagers ≥ 10*1048 Yes! Normal: (1048)(.66) ≈ 692 ≥ 10 and (1048)(.34) ≈ 356 ≥ 10 Yes! We are 95% confident that the true population proportion of teenagers with a TV in their room falls between .63 and .69.
Choosing the sample size Since the margin of error contains the sample proportion , we need to guess this value when choosing n . We will call this guess p*.
Choosing the sample size Two ways to get p * : 1. Use p* based on a past experience with similar studies. Cover several calculations to cover the range of -values you might find. Better to use when you have done a similar study.2. Use p* = 0.5 as the guess. The margin of error m is largest when . Use when you suspect to be between 0.3 and 0.7
Choosing the sample size So… Where p * is a guessed value for the sample proportion.
Example 12.9, p. 696 Gloria Chavez and Ronald Flynn are the candidates for mayor in a large city. You are planning a sample survey to determine what percent of the voters plan to vote for Chavez. This is a population proportion p . You will contact an SRS of registered voters in the city. You want to estimate p with 95% confidence and a margin of error no greater than 3%, or 0.03. How large a sample do you need?
Example 12.9, p. 696 Gloria Chavez and Ronald Flynn are the candidates for mayor in a large city. You are planning a sample survey to determine what percent of the voters plan to vote for Chavez. This is a population proportion p . You will contact an SRS of registered voters in the city. You want to estimate p with 95% confidence and a margin of error no greater than 3%, or 0.03. How large a sample do you need?Should we use p* = 0.5? YES!
Gloria Chavez and Ronald Flynn are the candidates for mayor in a large city. You are planning a sample survey to determine what percent of the voters plan to vote for Chavez. This is a population proportion p . You will contact an SRS of registered voters in the city. You want to estimate p with 95% confidence and a margin of error no greater than 3%, or 0.03. How large a sample do you need?So we want: 32.66 ≤ 1067.1≤ n So we need n = 1068 to satisfy this inequality.
Homework: p . 694: 12.8, 12.9 P. 696: 12. 10, 12.11 Due: Tuesday