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Section 8.3 A confidence interval to estimate one population mean. Review A point estimate is a single number that is our “best guess” for the parameter. ◦ The sample proportion is a point estimate of the population proportion. ◦ The sample mean is a point estimate of the population mean. Point Estimates A good estimator has two properties: ◦ The sampling distribution for the estimator is centered at the parameter of interest. Note that this is true for both the sampling distribution of the mean and proportion. ◦ The sampling distribution for the estimator has a relatively small standard error. Again, this is true for both mean & prop. Review An interval estimate is an interval of numbers within which the parameter is believed to fall. ◦ Because interval estimates contain the parameter with a certain degree of confidence these are often called confidence intervals. Confidence Intervals We build the confidence interval from the sampling distribution of the point estimator. We need an appropriately large sample size. ◦ For proportions: np > 15 & n(1 - p) > 15 ◦ For means: n > 30 Confidence Intervals The larger the confidence level, the wider the interval will be. The larger the sample size, the ______ the standard error will be. RECALL: Confidence Intervals for Proportions Use the sample proportion to estimate the standard error of the sampling distribution: s.e. ≈ pˆ (1 − pˆ ) n Determine how many standard errors to use by looking up the confidence level in the normal table to find the associated z. The confidence interval is then: ( pˆ − z * s.e., pˆ + z * s.e.) NEW: Confidence Intervals for Means Estimate a population mean µ using a confidence interval of the usual form: point estimate ± margin of error NEW: Confidence Intervals for Means Use the sample standard deviation to estimate the standard error of the sampling s distribution: s.e. ≈ n Because we are approximating the mean and the standard deviation in this case, the distribution is bell-shaped but not exactly normal. It actually has a t-distribution. The t-distribution has a bell-shape, gets more normal as n heads to infinity and has thicker tails than a normal distribution. Confidence Intervals for Means Determine how many standard errors to use by looking up the confidence level in the t-table to find the associated t. The confidence interval is then: (x − t * s.e., x + t * s.e.) The t*s.e. part is the margin of error. If we happen to know the value of σ we can use the normal distribution. If the population distribution is not normal the t-distribution is still robust. It performs adequately well in this case. Example Use a 95% CI to estimate the average number of hours per week that SU students work. ◦ Population? ◦ Categorical or quantitative variable? ◦ Parameter of interest? Example Suppose we randomly sample 40 SU students and find that the average number of hours that they work per week is 4.2 with a standard deviation of 2.8. Symbols: Point estimate: Standard error: Margin of error: CI Interval with Table B & with TI calculator. Confidence Intervals for Means Suppose that an experiment determines a 95% confidence interval for weight of a particular candy bar (in grams) of (8.1, 9.2). This means that we are 95% confident that the actual average weight of these candy bars is between 8.1 and 9.2 grams. Approximately 5% of the time a sample of candy bars will produce a confidence interval that doesn’t contain the true average weight.