Section 8.2 ~ Estimating Population Means Introduction to Probability and Statistics Ms. Young

Objective Sec. 8.2 After this section you will learn how to estimate population means and compute the associated margins of error and confidence intervals.

Confidence Interval Recall that a confidence interval is a range of values that is likely to contain the population parameter  Ex. ~ The population for a survey is all adults in the U.S. A sample of adults in the U.S. was taken and the people were asked how many hours per day they spend watching TV. The sample mean was found to be hours. If the margin of error for this was found to be.8 hours, what is the confidence interval? So based on this sample and the margin of error, we are 95% confident that adults in the U.S. watch between 1.5 hours and 3.1 hours of TV per day In other words, this range of values is 95% likely to contain the true population mean (which we would not know unless we took a census) There are other levels of confidence (90% and 99%), but we will be working with the 95% confidence level and how to calculate the margin of error using the sample mean Sec. 8.2

Margin of Error The margin of error for a 95% confidence interval is found using the following formula where s = the standard deviation from the sample and n = the sample size Once the margin of error is found, the confidence interval is written as follows:  Ex. ~ For the confidence interval found earlier (between 1.5 hours and 3.1 hours) it could be written as: Sec. 8.2 or

Example 1 Compute the margin of error and find the 95% confidence interval for the protein intake sample of n = 267 men, which has a sample mean of grams and a sample standard deviation of s = 58.6 grams. Sec. 8.2 The sample mean is grams, so the 95% confidence interval extends approximately from 77.0 – 7.2 = 69.8 grams to = 84.2 grams. We can conclude with 95% confidence, that the average protein intake of men is between 69.8 grams and 84.2 grams.

Choosing Sample Size Often times in statistical surveys and experiments, we know in advance the margin of error we would like to achieve  Ex. ~ We might want to estimate the mean cost of a new car within $200 If you know the margin of error that you would like to achieve, you can manipulate the formula to solve for the appropriate sample size needed to obtain that margin of error To find the exact sample size needed, you would use the population standard deviation, but often times you will not know this since you just study a sample  When using the population standard deviation to find the sample size, you always round the result up Sec. 8.2

Example 2 Sec. 8.2 You want to study housing costs in the country by sampling recent house sales in various regions. Your goal is to provide a 95% confidence interval estimate of the housing cost. Previous studies suggest that the population standard deviation is about $7,200. What sample size should be used to ensure that the sample mean is at least a. $500 of the true population mean? The sample should include at least 830 prices b. $100 of the true population mean? The sample should include at least 20,736 prices