6.1 Estimating with Confidence

How can we be sure? AGAIN, we look at this question about the mean of a sample estimating the mean of a population. This time, however, we will be able to say how “good” our mean sample is compared to the mean population, even though we don’t know what that is. We define a sample mean x to be a point estimate for the population μ.

But rather than giving a point estimate for the mean of the population, we seek to give an interval estimate; that is, an interval in which we think the population mean lies (obviously the smaller the better). We will also be able to give a level of confidence which answers the question ‘how sure are you?’ e.g. 95% sure, 98% sure etc. Let’s show how we might go about this in general on the board.

We have shown the following: For a sample size n>29, a 1-α confidence interval for a population mean μ is given by

Margin of error Notice in the above inequality that we have taken our point estimate and simply added z α/2 (σ/n 1/2 ) on the right side of the inequality and subtracted z α/2 (σ/n 1/2 ) on the left side of the inequality. We define the margin of error of the estimate with confidence 1-α to be m=z α/2 (σ/n 1/2 ).

Levels of Confidence The following table is helpful to have in front of you when computing confidence intervals: These and any other z values are easily found by looking at the table way in the back. Level of Confidence 1-α z α/2 90% % % %2.58

A physician wanted to estimate the mean length of time μ that a patient had to wait to see him after arriving at the office. A random sample of 50 patients showed a mean waiting time of 23.4 minutes and a standard deviation of 7.1 minutes. Find a 95% confidence interval for μ and identify the maximum error of estimate.

The mean monthly rent of an SRS of 44 unfurnished one bedroom apartments in Boston is $1400. Assume that the standard deviation is $220. Find a 95% confidence interval for the margin of error.

Practice Problem Suppose a student measuring the boiling temperature of a certain liquid observes readings (in degrees Celsius). He calculates the sample mean of 124 readings to be If he knows that the standard deviation for this procedure is 1.2 degrees, what is the confidence interval for the population mean at a 98% confidence level?

Determining the Sample Size Sometimes we may have a certain confidence interval in mind that we would like to be able to say our mean is within. In this case, we may ask what sample size we would need in order to fall within this interval. Let’s do this on the board. Hence, if we know the standard deviation and would like the mean to fall within a certain confidence interval, we must choose n at least

How many households in a large town should be randomly sampled to estimate the mean number of dollars spent per household per week on food supplies to within $3 with 80% confidence? Use σ=15.