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Statistics Lesson 8.1 Estimation µ when σ is Known Notes Page 1 of 4 Assumptions about the random variable x 1. We have a simple random sample of size n drawn from a population of x values. 2. The value of σ, the population standard deviation of x, is known 3. If the x distribution is normal, then our methods work for any sample size n. 4. If x has an unknown distribution then we require a sample size n ≥ 30. However, if the x distribution is distinctly skewed and definitely not moundshaped, or bell shaped, then a sample size of 50 or higher may be necessary. Point Estimate: A point estimate of a population parameter is an estimate of the parameter using a single number. o x is the point estimate for µ When using x as a point estimate for µ, the margin of error is the magnitude of x or x . Confidence level (c) The reliability of an estimate. For a confidence level c, the critical value zc is the number such that the area under the standard normal curve between – zc and zc equals c. The area under the normal curve from – zc and zc, is the probability that the standardized normal variable z lies in the interval. This means that P( zc z zc ) . Example 1: Find the critical value zc for c = 0.90. Example 2: Find P( z0.99 z z0.99 ) 0.99 . Statistics Lesson 8.1 Estimation µ when σ is Known Notes Page 2 of 4 c Confidence Interval for µ A c Confidence Interval for µ is an interval computed from sample data in such a way that c is the probability of generating an interval containing the actual value of µ. In other words, c is the proportion of confidence intervals, based on random samples of size n that actually contain µ. How to find a confidence interval for µ when σ is known… xE xE where x = sample mean of a simple random sample E zc n c = confidence level (0 < c < 1) zc = critical value for confidence level c based on the standard normal distribution. Example 3: Julia enjoys jogging. She has been jogging over a period of several years during which time her physical condition has remained constantly good. Usually, she jogs 2 miles per day. The standard deviation of her times is σ = 1.80 minutes. During the past year, Julia has recorded her times to run 2 miles. She has a random sample of 90 of these times. For these 90 times, the mean was x = 15.60 minutes. Let µ be the mean jogging time for the entire distribution of Julia’s 2 mile running times (taken over the past year.) Find a 0.95 confidence interval for µ. What does it mean? Statistics Lesson 8.1 Estimation µ when σ is Known Notes Page 3 of 4 Example 4: Walter usually meets Julia at the track. He prefers to jog 3 miles. From long experience, he knows that σ = 2.40 minutes for his jogging times. For a random sample of 90 sessions, Walter’s mean time was x = 22.50 minutes. Let µ be the mean jogging time for the entire distribution of Walter’s 3 mile running times over the past several years. Find a 0.99 confidence interval for µ. What does it mean? Example 5: Suppose that the standard deviation of all high school seniors’ SAT scores in a certain year was σ = 150. A random sample of 100 scores yielded the sample mean x = 1010. Let µ be the mean of all SAT scores in that year. Find a 0.99 confidence interval for µ. Round your answers to integers. Statistics Lesson 8.1 Estimation µ when σ is Known Notes Page 4 of 4 Example 6: Use the given information in Example 5. a. What is the value of z0.95 ? b. Is the x distribution approximately normal? c. What is the value E when c = 0.95? Round your answer to an integer. d. What are the endpoints for a 0.95 confidence interval for µ? Round your answers to integers. e. Interpret the confidence interval.