Download Notes on Confidence Intervals for the Mean | MATH 243 and more Study notes Probability and Statistics in PDF only on Docsity! MATH 243, LECTURE 15 1. Confidence intervals for the mean Today we continue to work on confidence intervals. The first case we will deal with is a population where we have some sense of its standard deviation (we will get around this restriction later). Example 1. It is a safe assumption that women’s heights have a deviation of less than five inches. Suppose some random sample of 400 women has an average height of 63.5 inches. Establish a 95% confidence interval for the true mean of women’s heights based on this sample. Now some formalism. Definition 2. The z∗ value for a confidence percentage C% is the number of standard deviations from the mean (both over and under) needed to account for C% of a normal distribution. You can look up z∗ values in Table C. Our computations above of confidence intervals are summarized as follows. Theorem 3. Suppose a sufficiently large SRS of population n is drawn from a population having an unknown mean but known (or at least bounded above) deviation σ. A confidence interval with confidence percentage C% is given by x± z∗ σ√ n , where z∗ is the critical value associated to C in Table C. Example 4. Suppose we wanted to estimate the mean property value in Eugene. It is reasonable to assume a standard deviation of less than 200K. Suppose we somehow surveyed 625 properties and found a mean of 284K. Establish a confidence interval with confidence 95% for the true mean. While we may prefer to work things through rather than use a formula, having the formula lets us understand the behavior of confidence intervals with respect to different parameters (all of which agree with common sense). • A high level of confidence requires a larger margin of error. • A small margin of error means a lower level of confidence. • The margin of error decreases as the sample size increases. • A large initial deviation increases the margin of error. 2. Finding sufficiently large populations Most often the initial deviation is set, as is our desired margin of error and level of confidence; that leaves us solving for which population will allow us to achieve this confidence interval with the parameters given. We can be pretty loose in estimating standard deviation. Example 5. Suppose we assume that the standard deviation for IQ is less than 35 (a reasonable assump- tion). How many IQ’s would we need to randomly survey to find the average IQ within a confidence interval of 1 with 95% certainty? Example 6. Assuming American men’s heights have a deviation of at most five inches, how many men must be measured to determine the average height within one tenth of an inch with 90% confidence. 1
