Unit 5: Inference for categorical variables Lecture 1: Inference for proportions Statistics 101 Thomas Leininger June 12, 2013
Single population proportion Single population proportion 1 Identifying when a sample proportion is nearly normal Confidence intervals for a proportion Choosing a sample size when estimating a proportion Hypothesis testing for a proportion Small sample inference for a proportion 2 Carnival Game Paul the octopus Statistics 101 U5 - L1: Inf. for proportions Thomas Leininger
Single population proportion Many research questions involve proportions Who will win the election? http://elections.huffingtonpost.com/2012/results Statistics 101 (Thomas Leininger) U5 - L1: Inf. for proportions June 12, 2013 2 / 23
Single population proportion Many research questions involve proportions Who will win the NBA finals? Statistics 101 (Thomas Leininger) U5 - L1: Inf. for proportions June 12, 2013 2 / 23
Single population proportion Many research questions involve proportions Mac or PC? Statistics 101 (Thomas Leininger) U5 - L1: Inf. for proportions June 12, 2013 2 / 23
Single population proportion Many research questions involve proportions Is this the cutest baby in the world? Statistics 101 (Thomas Leininger) U5 - L1: Inf. for proportions June 12, 2013 2 / 23
Single population proportion Question Two scientists want to know if a certain drug is effective against high blood pressure. The first scientist wants to give the drug to 1000 peo- ple with high blood pressure and see how many of them experience lower blood pressure levels. The second scientist wants to give the drug to 500 people with high blood pressure, and not give the drug to another 500 people with high blood pressure, and see how many in both groups experience lower blood pressure levels. Which is the better way to test this drug? (a) All 1000 get the drug (b) 500 get the drug, 500 don’t Statistics 101 (Thomas Leininger) U5 - L1: Inf. for proportions June 12, 2013 3 / 23
Single population proportion Question Two scientists want to know if a certain drug is effective against high blood pressure. The first scientist wants to give the drug to 1000 peo- ple with high blood pressure and see how many of them experience lower blood pressure levels. The second scientist wants to give the drug to 500 people with high blood pressure, and not give the drug to another 500 people with high blood pressure, and see how many in both groups experience lower blood pressure levels. Which is the better way to test this drug? (a) All 1000 get the drug (b) 500 get the drug, 500 don’t Statistics 101 (Thomas Leininger) U5 - L1: Inf. for proportions June 12, 2013 3 / 23
Single population proportion Results from the GSS The GSS asks the same question, below is the distribution of responses from the 2010 survey: All 1000 get the drug 99 500 get the drug 500 don’t 571 Total 670 Statistics 101 (Thomas Leininger) U5 - L1: Inf. for proportions June 12, 2013 4 / 23
Single population proportion Parameter and point estimate We would like to estimate the proportion of all Americans who have a good intuition about experimental design, i.e. would answer “500 get the drug 500 don’t?” What are the parameter of interest and the point estimate? Statistics 101 (Thomas Leininger) U5 - L1: Inf. for proportions June 12, 2013 5 / 23
Single population proportion Parameter and point estimate We would like to estimate the proportion of all Americans who have a good intuition about experimental design, i.e. would answer “500 get the drug 500 don’t?” What are the parameter of interest and the point estimate? Parameter of interest: Proportion of all Americans who have a good intuition about experimental design. p ( a population proportion ) Statistics 101 (Thomas Leininger) U5 - L1: Inf. for proportions June 12, 2013 5 / 23
Single population proportion Parameter and point estimate We would like to estimate the proportion of all Americans who have a good intuition about experimental design, i.e. would answer “500 get the drug 500 don’t?” What are the parameter of interest and the point estimate? Parameter of interest: Proportion of all Americans who have a good intuition about experimental design. p ( a population proportion ) Point estimate: Proportion of sampled Americans who have a good intuition about experimental design. p ( a sample proportion ) = 571 / 670 = 0 . 85 ˆ Statistics 101 (Thomas Leininger) U5 - L1: Inf. for proportions June 12, 2013 5 / 23
Single population proportion Inference on a proportion What percent of all Americans have a good intuition about experimen- tal design, i.e. would answer “500 get the drug 500 don’t?” Statistics 101 (Thomas Leininger) U5 - L1: Inf. for proportions June 12, 2013 6 / 23
Single population proportion Inference on a proportion What percent of all Americans have a good intuition about experimen- tal design, i.e. would answer “500 get the drug 500 don’t?” We can answer this research question using a confidence interval, which we know is always of the form point estimate ± ME Statistics 101 (Thomas Leininger) U5 - L1: Inf. for proportions June 12, 2013 6 / 23
Single population proportion Inference on a proportion What percent of all Americans have a good intuition about experimen- tal design, i.e. would answer “500 get the drug 500 don’t?” We can answer this research question using a confidence interval, which we know is always of the form point estimate ± ME And we also know that ME = critical value × standard error of the point estimate. SE ˆ p = ? Statistics 101 (Thomas Leininger) U5 - L1: Inf. for proportions June 12, 2013 6 / 23
Single population proportion Inference on a proportion What percent of all Americans have a good intuition about experimen- tal design, i.e. would answer “500 get the drug 500 don’t?” We can answer this research question using a confidence interval, which we know is always of the form point estimate ± ME And we also know that ME = critical value × standard error of the point estimate. SE ˆ p = ? Standard error of a sample proportion � p (1 − p ) SE ˆ p = n Statistics 101 (Thomas Leininger) U5 - L1: Inf. for proportions June 12, 2013 6 / 23
Single population proportion Identifying when a sample proportion is nearly normal Single population proportion 1 Identifying when a sample proportion is nearly normal Confidence intervals for a proportion Choosing a sample size when estimating a proportion Hypothesis testing for a proportion Small sample inference for a proportion 2 Carnival Game Paul the octopus Statistics 101 U5 - L1: Inf. for proportions Thomas Leininger
Single population proportion Identifying when a sample proportion is nearly normal Sample proportions are also nearly normally distributed Central limit theorem for proportions Sample proportions will be nearly normally distributed with mean equal � p (1 − p ) to the population mean, p , and standard error equal to . n � p (1 − p ) p ∼ N ˆ mean = p , SE = n Statistics 101 (Thomas Leininger) U5 - L1: Inf. for proportions June 12, 2013 7 / 23
Single population proportion Identifying when a sample proportion is nearly normal Sample proportions are also nearly normally distributed Central limit theorem for proportions Sample proportions will be nearly normally distributed with mean equal � p (1 − p ) to the population mean, p , and standard error equal to . n � p (1 − p ) p ∼ N ˆ mean = p , SE = n But of course this is true only under certain conditions... any guesses? Statistics 101 (Thomas Leininger) U5 - L1: Inf. for proportions June 12, 2013 7 / 23
Single population proportion Identifying when a sample proportion is nearly normal Sample proportions are also nearly normally distributed Central limit theorem for proportions Sample proportions will be nearly normally distributed with mean equal � p (1 − p ) to the population mean, p , and standard error equal to . n � p (1 − p ) p ∼ N ˆ mean = p , SE = n But of course this is true only under certain conditions... any guesses? independent observations, at least 10 successes and 10 failures Statistics 101 (Thomas Leininger) U5 - L1: Inf. for proportions June 12, 2013 7 / 23
Single population proportion Identifying when a sample proportion is nearly normal Sample proportions are also nearly normally distributed Central limit theorem for proportions Sample proportions will be nearly normally distributed with mean equal � p (1 − p ) to the population mean, p , and standard error equal to . n � p (1 − p ) p ∼ N ˆ mean = p , SE = n But of course this is true only under certain conditions... any guesses? independent observations, at least 10 successes and 10 failures Note: If p is unknown (most cases), we use ˆ p when doing a CI and p 0 when doing a HT. Statistics 101 (Thomas Leininger) U5 - L1: Inf. for proportions June 12, 2013 7 / 23
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