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Download Confidence Intervals for the Mean • c is the confidence level (usually
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Confidence Intervals for the Mean • c is the confidence level (usually .9, .95 or .99) but it can be any value between 0 and q. • the critical value zc is the number such that the area under the standard normal curve between −zc and zc equals c. • Given a sample of size n from a population (if population is not normal n must be at least 30), with sample mean x̄, there is a probability of c that σ σ x̄ − zc √ < µ < x̄ + zc √ n n This is the confidence interval for µ at confidence level c. If σ is not known, the sample standard deviation is often used in place of σ along with the student’s t-distribution. This is discussed in Section 8.2. σ • For a given confidence level c, the quantity E = zc √ is referred to as the maximal margin n of error. Example 1. Find zc for (a) c = .99, (b) c = .95 and (c) c = .92. Example 2. (Section 8.1#4) A random sample of 40 farmers gave a sample mean of x̄ = $6.88 received by farmers per 100 pound of watermelon. Assume that σ is known to be $1.92 per 100 pounds. (a) Find a 90% confidence interval for the population mean price (per 100lbs) that farmers get for their watermelon crop. What is the margin of error? Repeat this question, except with a 99% confidence interval. Do you expect a longer or shorter interval for a higher level of confidence? (b) Find the 90% confidence interval for the mean amount of money farmers will receive for 15 tons of watermelon. What is the margin of error? Example 3. Suppose you wish to find the average annual salary for school teachers in the United States. Suppose you took a random sample and found that x̄ = 53, 145 and you estimated σ = $16, 451. Find a 95% confidence interval for the mean salary, given that your sample had size (a) n = 36, (b) n = 64, (c) n = 144. (d) Does the margin of error increase or decrease as n gets larger. What effect did quadrupling n have on the margin of error? (e) Does the confidence interval get longer or shorter as the sample size increases? Is that what you would expect?
