Chapter 8, Section 8.2: Estimating a Population Mean - σ Not Known

Chapter 8 – Confidence Intervals – One Population Section 8.2 - Estimating a Population Mean: σ Not Known Conditions for Using the Student t Distribution • σ is unknown, (if σ is known we use z); • The sample is a simple random sample • Either or both of these conditions is satisfied: i) The population is normally distributed, or ii) n ≥ 30 (The sample has 30 or more values) Confidence Interval for the Estimate of μ (With σ Not Known) 1. Verify that the required assumptions are satisfied. 2. Find the critical value tc from table 6 (n -1 degrees of freedom). 3. Evaluate the margin of error E. (E = tc s n ) 4. Then using E and the sample mean the confidence interval is: x−E  x+E or ( x − E, x + E ) Using the TI-83 to Construct a Confidence Interval for Estimating μ, (σ Not Known) STAT>>TESTS select 8:TInterval Round-off Rule for Confidence Intervals used to Estimate μ: a) If original data is given: use one more decimal place than original values. b) If you are given summary statistics from a data set, use the same number of decimal places used for the sample mean. QUESTION 3 - Under the same conditions of SAME SAMPLE SIZE, and SAME CONFIDENCE, which interval is shorter (more precise): the z-interval or the t-interval? Comparing (2-a) and (1-a) will help you in answering this question 2) A simple random sample of size n is drawn from a population. If that sample has a mean of 59.2 and a standard deviation of 3.8, a) Compute the 90% confidence interval about µ if the sample size, n, is 45. b) Compare with your answer to part 3-a. Which interval is longer? Which interval is more precise? Finding the Point Estimate and the Margin of Error from a Confidence Interval 3) We are 95% confident that the interval from 98.08°F to 98.32°F actually contains the true mean value of the body temperatures of all healthy adults. What is the point estimate in this case? What is the margin of error? 4) In order to correctly diagnose the disorder of hydrocephalus, a pediatrician investigates head circumferences of two month old babies. Use the sample MHED, augment it with the sample FHED. This will give you a new sample of size 100. Use this sample to estimate the head circumference of all two months old babies. Notice: Spring 09 students. I have not given you these lists, so here are the statistics that you need to use in this n = 100 problem s = 1.649 x = 40.573 a) What is the point estimate? b) Verify that the requirements for constructing a confidence interval about x-bar are satisfied. c) Construct a 99% confidence interval estimate for the head circumference of all two months old babies. (Are you using z or t? Why?) d) The statement “99% confident” means that, if 100 samples of size _____ were taken, about _____ intervals will contain the parameter μ and about ____ will not. e) Complete the following: We are _____% confident that the mean head circumference of all two months old babies is between _____ and ______ f) With ______% confidence we can say that the mean head circumference of all two months old babies is ______ with a margin of error of _______ g) For ______% of intervals constructed with this method, the sample mean would not differ from the actual population mean by more than ______ h) How can you produce a more precise confidence interval? i) How large of a sample should be selected in order to be 99% confident that the point estimate x-bar will be within 0.2 cm of the true population mean?