Inference on Proportions Assumptions: • SRS • Normal distribution np > 10 & n(1-p) > 10 • Population is at least 10n Formula for Confidence interval: CI statistic critical valueSD of statistic Normal curve p 1 p ˆ p z* n Note: For confidence intervals, we DO NOT know p – so we MUST substitute p-hat for p in both the SD & when checking assumptions. A May 2000 Gallup Poll found that 38% of a random sample of 1012 adults said that they believe in ghosts. Find a 95% confidence interval for the true proportion of adults who believe in ghost. Assumptions: Step 1: check assumptions! •Have an SRS of adults •np =1012(.38) = 384.56 & n(1-p) = 1012(.62) = 627.44 Since both are greater than 10, the distribution can be approximated by a normal curve Step 2: make •Population of adults is at least 10,1012.calculations ˆ z * p 1 p .38 1.96 .38(.62) .35,.41 P n 1012 Step 3: conclusion in context We are 95% confident that the true proportion of adults who believe in ghosts is between 35% and 41%. Another Gallop Poll is taken in order to measure the To find sample size: 1 p proportion of adultspwho m z * approve of attempts to clonen However, since we have not yet is humans. What sample size taken a sample, to be within p-hat (or of the necessarywe do not know a + 0.04 p) to use! true proportion of adults who approve of attempts to clone humans with a 95% Confidence Interval? What p-hat (p) do you use when trying to find the sample size for a given margin of error? .1(.9) = .09 By using .5 for p-hat, .2(.8) = .16 we are using the worst- .3(.7) = .21 case scenario and using the largest SD in .4(.6) = .24 our calculations. .5(.5) = .25 Another Gallop Poll is taken in order to measure the proportion of adults who approve of attempts to clone humans. What sample size is necessary to be within + 0.04 of the true proportion of adults who approve of attempts to clone humans with a 95% Confidence Interval? p 1 p m z * n .5.5 Use p-hat = .5 .04 1.96 n .04 .5.5 Divide by 1.96 1.96 n 2 .04 .25 Square both sides 1.96 n n 600 .25 601 Round up on sample size Stop & do homework! Hypotheses for proportions: H0: p = value Ha: p > value Use >, <, or ≠ where p is the true proportion of context Formula for hypothesis test: statistic - parameter Test statistic SD of statistic ˆ p p z p 1 p n A company is willing to renew its advertising contract with a local radio station only if the station can prove that more than 20% of the residents of the city have heard the ad and recognize the company’s product. The radio station conducts a random sample of 400 people and finds that 90 have heard the ad and recognize the product. Is this sufficient evidence for the company to renew its contract? Assumptions: •Have an SRS of people •np = 400(.2) = 80 & n(1-p) = 400(.8) = 320 - Since both are greater than 10, this distribution is approximately normal. •Population of people is at least 4000. Use the parameter in the null H0: p = .2 where p is the hypothesis to check assumptions! true proportion of people who Ha: p > .2 heard the ad .225 .2 z 1.25 p value .1056 α .05 .2(.8) Use the parameter in the null 400 hypothesis to calculate standard deviation! Since the p-value >a, I fail to reject the null hypothesis. There is not sufficient evidence to suggest that the true proportion of people who heard the ad is greater than .2.
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