Download Approximating Sampling Distributions & Confidence Intervals: Understanding p & Properties and more Study notes Data Analysis & Statistical Methods in PDF only on Docsity! 16 September How to approximate a sampling distribution for p Decide on a sample size n Randomly select a sample of size n from the population Compare the proportion of successes, p Repeat steps 2 and 3 ≥ 1000 times. Plot a histogram of the p values to see what the sampling distribution looks like. Properties for the distribution of p Let p be the proportion of successes in a random sample of size n from a population whose proportion of successes is π. Denote the mean value of p’s distribution (thus, of p) by μp, and the standard deviation of p by σp. We know μp = π σp = √ π (1−π )n If it is large and π is not extreme (i.e., π is not close to 0 or 1), then the distribution of p is approximately normal. Rule: Normal if both πn ≥ 10 and (1-π)n ≥ 10 Probabilities for p Say we want to find P(p < .34) We can find this since P~N(μp, σ2p) when rule holds. So we standardize p using a Z-score where Z = p−μpp σpp = p−π √ π (1−π )n And use standard normal tables to solve. Standard error The standard error is an estimate of the standard deviation. The standard error for p, denoted SE(p) is SE(p) = √ p(1−p)n Why do we use the standard curve? The estimates we get from the samples are incorrect. We can only hope that the estimates are close to the true values. Standard errors help us determine a range of possible values that may contain the true value. These ranges are called confidence intervals. Estimation using a single sample Point estimates A point estimate of a population characteristic is a single number that is based on sample data and represents a plausible value of the population characteristic. i.e. p is a point estimate for π Standard error is a point estimate for standard deviation x is a point estimate for μ There can be different point estimates for one population characteristic x and median can both estimate μ We want point estimates to be unbiased. Its mean value is equal to the population parameter. p is unbiased, since μp = π If we have multiple (or no) unbiased point estimates, we choose the point estimate with the smallest variance Interval Estimation Let Δ be a fixed, but unknown parameter value Let δ be the point estimate for Δ
