Estimating a Population Mean Warmup You work for a consumer advocate agency and want to find the mean repair cost of a washing machine. As part of your study, you randomly select 40 repair costs and find the mean to be $100.00. From past studies, you assume that the σ is $17.50. Construct a 90% confidence interval. (Include all four steps: state the problem, check conditions, calculate the CI, and interpret it). Estimating a Population Mean

Section 8.3 Estimating a Population Mean Learning Objectives After this section, you should be able to… CONSTRUCT and INTERPRET a confidence interval for a population mean DETERMINE the sample size required to obtain a level C confidence interval for a population mean with a specified margin of error DESCRIBE how the margin of error of a confidence interval changes with the sample size and the level of confidence C DETERMINE sample statistics from a confidence interval

Estimating a Population Mean The One-Sample z Interval for a Population Mean To calculate a confidence interval for µ , we use the familiar formula: estimate ± (critical value) • (standard deviation of statistic) Estimating a Population Mean Choose an SRS of size n from a population having unknown mean µ and known standard deviation σ. As long as the Normal and Independent conditions are met, a level C confidence interval for µ is The critical value z* is found from the standard Normal distribution. One-Sample z Interval for a Population Mean

Estimating a Population Mean Example #1 Researchers would like to estimate the mean cholesterol level µ of a particular variety of monkey that is often used in laboratory experiments. The standard deviation of cholesterol level is 5 mg/dl. A random sample of 40 monkeys was used and the sample mean was 53 mg/dl. Construct a 98% confidence interval for the mean cholesterol level of this type of monkey. Estimating a Population Mean

Estimating a Population Mean Example #2 Assume that systolic blood pressure (SBP) for healthy adults is normally distributed with σ = 20 mm Hg. If an SRS of 20 adults is selected and the mean of the sample is 118 mm HG, construct a 99% confidence interval for the mean systolic blood pressure of all adults. Estimating a Population Mean

Estimating a Population Mean Choosing the Sample Size The margin of error ME of the confidence interval for the population mean µ is Estimating a Population Mean We determine a sample size for a desired margin of error when estimating a mean in much the same way we did when estimating a proportion. Choosing Sample Size for a Desired Margin of Error When Estimating µ To determine the sample size n that will yield a level C confidence interval for a population mean with a specified margin of error ME: • • Set the expression for the margin of error to be less than or equal to ME and solve for n:

Estimating a Population Mean Example: How Many Monkeys? Researchers would like to estimate the mean cholesterol level µ of a particular variety of monkey that is often used in laboratory experiments. They would like their estimate to be within 1 milligram per deciliter (mg/dl) of the true value of µ at a 95% confidence level. A previous study involving this variety of monkey suggests that the standard deviation of cholesterol level is about 5 mg/dl. Estimating a Population Mean

Estimating a Population Mean Example: Blood Pressure Assume that systolic blood pressure (SBP) for healthy adults is normally distributed with σ = 20 mm Hg. What sample size is needed so that 95% of sample means are between 116 mmHg and 124 mmHg? Estimating a Population Mean