Confidence intervals for a population mean*
The standard error of ˉx depends on both the sample size n and the standard deviation σ of individuals in the population. We know n but not σ. When n is large, the sample standard deviation s is close to σ and can be used to estimate it, just as we use the sample mean ˉx to estimate the population mean μ. The estimated standard error of ˉx is, therefore, s/√n. Now we can find confidence intervals for μ following the same reasoning that led us to confidence intervals for a proportion p. The big idea is that to cover the central area C under a Normal curve, we must go out a distance z* on either side of the mean. Look again at Figure 21.5 to see how C and z* are related.
Confidence interval for a population mean
Choose an SRS of size n from a large population of individuals having mean μ. The mean of the sample observations is ˉx. When n is reasonably large, an approximate level C confidence interval for μ is
ˉx±z*s√n
where z* is the critical value for confidence level C from Table 21.1.
The cautions we noted in estimating p apply here as well. The recipe is valid only when an SRS is drawn and the sample size n is reasonably large. How large is reasonably large? The answer depends upon the true shape of the population distribution. A sample size of n ≥ 15 is usually adequate unless there are extreme outliers or strong skewness. For clearly skewed distributions, a sample size of n ≥ 40 often suffices if there are no outliers.
The margin of error again decreases only at a rate proportional to √n as the sample size n increases. And it bears repeating that ˉx and s are strongly influenced by outliers. Inference using ˉx and s is suspect when outliers are present. Always look at your data.
EXAMPLE 7 NAEP quantitative scores
The National Assessment of Educational Progress (NAEP) includes a mathematics test for high school seniors. Scores on the test range from 0 to 300. Demonstrating the ability to use the Pythagorean theorem to determine the length of a hypotenuse is an example of the skills and knowledge associated with performance at the Basic level. An example of the knowledge and skills associated with the Proficient level is using trigonometric ratios to determine length.
In 2009, 51,000 12th-graders were in the NAEP sample for the mathematics test. The mean mathematics score was ˉx=153, and the standard deviation of their scores was s = 34. Assume that these 51,000 students were a random sample from the population of all 12th-graders. On the basis of this sample, what can we say about the mean score μ in the population of all 12th-grade students?
The 95% confidence interval for μ uses the critical value z* = 1.96 from Table 21.1. The interval is
ˉx±z*s√n=153±1.9634√51,000
= 153 ± (1.96)(0.151) = 153 ± 0.3
We are 95% confident that the mean score for all 12th-grade students lies between 152.7 and 153.3.
NOW IT’S YOUR TURN
Question 21.3
21.3 Blood pressure of executives. The medical director of a large company looks at the medical records of 72 executives between the ages of 35 and 44 years. He finds that the mean systolic blood pressure in this sample is ˉx=126.1 and the standard deviation is s = 15.2. Assuming the sample is a random sample of all executives in the company, find a 95% confidence interval for μ, the unknown mean blood pressure of all executives in the company.
21.3 The 95% confidence interval for μ uses the critical value z* = 1.96 from Table 21.1. The interval is
ˉx±z* s√n=126.1±1.96 15.2√72
= 126.1 ± 1.96(1.79)
= 126.1 ± 3.5
We are 95% confident that the mean blood pressure for all executives in the company between the ages of 35 and 44 lies between 122.6 and 129.6.
*This section is optional.