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Unformatted text preview: 1 STA2023 – Chapter 23 (Inference About a Population Mean μ ) Skip: “Sample Size” (pg. 602-603) Chapters 19-22 introduced you to making inferences (confidence intervals & hypothesis testing) for proportion data – including inferences for a single population proportion ( p ) as well as comparing two populations proportions ( p 1 versus p 2 ) This chapter introduces inferences for a single population mean ( μ ). You should see that the approach to generating a confidence interval for μ is very much like what we did when calculating a confidence interval for p , and the methodology for running a hypothesis test about μ is very much like the methodology of a hypothesis test about p . Activity 1: It seems that the cost of college textbooks has become increasingly costly. Here are the costs of a random sample of 25 college-level textbooks, rounded to the nearest dollar. 122 166 171 148 135 173 137 163 119 144 164 153 162 140 142 158 130 167 173 186 92 170 126 163 172 A. Suppose we want to estimate the average cost of a college-level textbook. In other words, we want to estimate μ . Not knowing the true value of μ , what could we use as a point estimate of μ ? B. The point estimate from above ( _ y = $151.04) is based on only one sample of size n = 25. Because of , another sample of size n = 25 would likely give a different point estimate. Because the sample mean of $151.04 could be either an overestimate or an underestimate, we should generate a confidence interval for the true average cost ( μ ) per book for all college-level textbooks. Recall that a confidence interval is always of the form: (point estimate) ± (margin of error) The Central Limit Theorem (from chapter 18b) helps us determine what to use for the margin of error: When the sample size ( n ) is large, the distribution of sample means ( _ y-values) taken from the population can be approximated with a Normal model, where: _ y μ = μ _ y σ = / n σ Recall that this approximation is true regardless of whether the population’s distribution is symmetric, skewed, unimodal, bimodal, etc. Furthermore, the approximation gets better as n increases. So, assuming a Normal approximation, the margin of error for estimating μ is: ME = _ * ( ) z SD y ⋅ = * ( / ) z n σ ⋅ Thus, the resulting confidence interval formula for estimating μ is: _ y ± * z n σ ⋅ (called a 1-Sample z-Interval ) But…there is a problem. We don’t know σ (the standard deviation of the entire population). What can we use in place of σ ? 2 By estimating σ with s , we are introducing extra variability into our calculations, which must be accounted for in the calculation of the margin of error…and the distribution of _ y-values can no longer be approximated with a Normal model. That’s the bad news…the good news is that the distribution of _ y...
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This note was uploaded on 11/22/2010 for the course STA 2023 taught by Professor Ripol during the Spring '08 term at University of Florida.
- Spring '08