Statistical Inference for Population Proportions - Prof. Corlis E. Robe, Exams of Statistics

Download Statistical Inference for Population Proportions - Prof. Corlis E. Robe and more Exams Statistics in PDF only on Docsity! ROBE Math 1530 Spring 2005 Ch. 18, p. 469 “Inference about a population proportion” What proportion of voters really voted for W. in the 2004 presidential election? This wants a confidence interval for the proportion of W. votes. Is the proportion of students who fail Math 1530 really only 12% as Professor Robe claims? We need to run a hypothesis test for the proportion of Math 1530 students who fail. H0 : p = 12%, Ha : p > 12%. A population proportion p is a parameter just like a population mean . Example p. 250 #2. Just like a sample mean x estimates a population mean , a sample proportion p̂ estimates a population proportion p. p. 470 p̂ = count of successes in a sample / sample size n example p. 470 #1. If the study of religious practices were repeated – would they get those same 127 students in the sample? Would they get the same count 107 who pray? Sooo - p̂ is a variable. A variable from a random sample is a random variable, so probability applies, if we ask the question right. Across lots and lots of samples, we know that x will balance out around μ – the mean of x ’s sampling distribution is μ – and we know what the distribution of those x ’s will be. What about p̂ ? Box p. 471 (plus some) Choose an SRS of size n from a large population that contains a proportion p of ‘successes.’ Let p̂ be the sample proportion of successes p̂ = count of successes / n . Δ As n increases, the sampling distribution of p̂ becomes approximately Normal – The rule of thumb is that both np and n(1-p) > 10; most statisticians prefer that we also have n > 400 Δ μ ( p̂ ) = p provided the samples are SRS’s. Δ σ ( p̂ ) = n )p(p 1 provided n is less than 10% of the population. Example p. 472 #3a) b) Given a population’s proportion p, σ ( p̂ ) = n )p(p 1 is the standard deviation of p̂ , i.e., the standard deviation of p̂ ’s sampling distribution. In the event that the actual p is not available, we estimate with the sample’s proportion p̂ and n )p̂(p̂ 1 = SE ( p̂ ) is called the standard error of p̂ (p. 474) Example p. 470 #1 p is the proportion of college students who pray, p̂ = 107/127 = 0.84 and SE( p̂ ) =