6.1 Confidence Intervals for the Mean (Large Samples) Prob & Stats Mrs. O’Toole

Objectives Students will learn how to: Find a point estimate and a margin of error Construct and interpret confidence intervals for the population mean Determine the minimum sample size required when estimating the mean

Definition A point estimate is a single value estimate for a population parameter. The most unbiased point estimate of the population mean μ is the sample mean, Example: Complete “Try it yourself 1” on page 310 of your textbook. Solution: sample mean = 14.8 The mean number of sentences per magazine ad is 14.8

Definition In the previous example, the probability that the population mean is exactly 14.8 is virtually zero. So, instead of estimating μ to be exactly 14.8, it makes sense to estimate that μ lies within an interval. An interval estimate is an interval, or range of values, used to estimate a population parameter Example: Guessing your age

Definition The level of confidence c is the probability that the interval estimate contains the population parameter See the bottom of page 311 in your text. Draw the normal curve here: (See the green box on page 311, as well)

Definition The difference between the point estimate and the actual parameter is called the sampling error. You can calculate a maximum value for the error if you know the level of confidence and the sampling distribution.

Definition Given a level of confidence, c, the margin of error, E, is the greatest possible distance between the point estimate and the value of the parameter it is estimating. In order to use this technique, it is assumed that the population standard deviation is known. This is rarely the case, but when n  30, the sample standard deviation can be used in place of the population standard deviation.

Example Finding the margin of error As a class, complete Example 2 on page 312 of your text. With a partner, complete “Try it yourself 2” on page 312 of your text. Solution: z c = 1.96, n = 30, s  16.5, E = 5.9 You are 95% confident that the maximum error of the estimate is about 5.9 sentences per magazine ad.

Definition A c-confidence interval for the population mean is: The probability that the confidence interval contains μ is c.

Guidelines Finding a confidence interval 1. Find the sample statistics n and x-bar. 2. Specify σ, if known. Otherwise, if n  30, find the sample standard deviation s. 3. Find the critical value z c that corresponds to the given level of confidence. 4. Find the margin of error, E. 5. Find the left and right endpoints of the confidence interval. (See page 313 in your text for the formulas)

Examples With a partner, complete “Try it yourself 3” and “Try it yourself 5” in your notes. Solutions: Try it yourself 3 x-bar = 14.8, E = 5.9, left 8.9, right 20.7 Try it yourself 5 n = 30, x-bar = 22.9, σ = 1.5, z c = 1.645, E = 0.5, left 22.4, right 23.4

A note about sample size As level of confidence increases, the confidence interval widens. As the confidence interval widens, the precision of the estimate decreases. One way to improve the precision of the estimate without decreasing the level of confidence is to increase the sample size. Minimum sample size needed to estimate μ:

Example Determining minimum sample size Complete “Try it yourself 6” on page 316 of your text. Solution: z c = 1.96, E = 2, s ≈ 5.0, n = 25

Homework 6.1 p.317 (1-5, 9-17 odd, odd, 38, 56)