Review the Learning Activity titled “Large Sample Estimation of a Population Mean.” Explain how Figure 4.46 depicts the title associated with it. In your own words, what does it mean to construct a 95% confidence interval? What would you say to a student who thinks that it means “95% of your prediction is accurate while the other 5% is in error?”
If we wish to estimate the mean μ of a population for which a census is impractical, say the average height of all 18-year-old men in the country, a reasonable strategy is to take a sample, compute its mean , and estimate the unknown number μ by the known number . For example, if the average height of 100 randomly selected men aged 18 is 70.6 inches, then we would say that the average height of all 18-year-old men is (at least approximately) 70.6 inches. Estimating a population parameter by a single number like this is called point estimation; in the case at hand, the statistic is a point estimate of the parameter μ. The terminology arises because a single number corresponds to a single point on the number line. A problem with a point estimate is that it gives no indication of how reliable the estimate is. In contrast, in this section we learn about interval estimation. If we collected an infinite number of samples of the population of all 18-year-old men, and computed a sample mean of their heights and corresponding confidence interval, what percentage of those intervals would contain the true average height of all 18-year-old men? In brief, in the case of estimating a population mean μ, we use a formula to compute from the data a number E, called the margin of error of the estimate, and form the interval We do this in such a way that a certain proportion, say 95%, of all the intervals constructed from sample data by means of this formula contain the unknown parameter μ. Such an interval is called a 95% confidence interval for μ. Continuing with the example of the average height of 18-year-old men, suppose that the sample of 100 men mentioned previously for which inches also had sample standard deviation s = 1.7 inches. It then turns out that E = 0.33, and we would state that we are 95% confident that the average height of all 18-year-old men is in the interval formed by inches, that is, the average is between 70.27 and 70.93 inches. If the sample statistics had come from a smaller sample, say a sample of 50 men, the lower reliability would show up in the 95% confidence interval being longer, hence, less precise in its estimate. In this example, the 95% confidence interval for the same sample statistics but with n = 50 is inches, or from 70.13 to 71.07 inches. Large Sample Estimation of a Population Mean The central limit theorem says that, for large samples (samples of size n ≥ 30), when viewed as a random variable, the sample mean is normally distributed with mean and standard deviation . The empirical rule says that we must go about two standard deviations from the mean to capture 95% of the values of generated by sample after sample. A more precise distance based on the normality of is 1.960 standard deviations, which is The key idea in the construction of the 95% confidence interval is this: in sample after sample, 95% of the values of lie in the interval , if we adjoin to each side of the point estimate a “wing” of length E, 95% of the intervals formed by the winged dots contain μ. The 95% confidence interval is . For a different level of confidence, say 90% or 99%, the number 1.960 will change, but the idea is the same. Figure 4.46 When Winged Dots Capture the Population Mean The next figure shows the intervals generated by a computer simulation of drawing 40 samples from a normally distributed population and constructing the 95% confidence interval for each one. We expect that about (0.05)(40) = 2 of the intervals so constructed would fail to contain the population mean μ, and in this simulation, two of the intervals, shown in red, do. Figure 4.47 Computer Simulation of 40 95% Confidence Intervals for a Mean It is standard practice to identify the level of confidence in terms of the area in the two tails of the distribution of when the middle part specified by the level of confidence is taken out. Remember that the z-value that cuts off a right tail of area c is denoted zc. Thus the number 1.960 in the example is , which is for Figure 4.48 For % confidence the area in each tail is Figure 4.49 Area in each tail For 95% confidence, the area in each tail is The level of confidence can be any number between 0 and 100%, but the most common values are probably 90% (), 95% (), and 99% (). Thus, in general, for a % confidence interval, , so the formula for the confidence interval is . While sometimes the population standard deviation σ is known, typically it is not. If not, for n ≥ 30, it is generally safe to approximate σ by the sample standard deviation s. Large Sample Confidence Interval for a Population Mean If σ is known: If σ is unknown: A sample is considered large when n ≥ 30. As mentioned earlier, the number is called the margin of error of the estimate. Example 1 Find the number needed in construction of a confidence interval: 1. when the level of confidence is 90% 2. when the level of confidence is 99% Solution (a) For confidence level 90%, , so Because the area under the standard normal curve to the right of is 0.05, the area to the left of z0.05 is 0.95. We search for the area 0.9500 in the cumulative probability table. The closest entries in the table are 0.9495 and 0.9505, corresponding to z-values 1.64 and 1.65. Since 0.95 is exactly halfway between 0.9495 and 0.9505 we use the average 1.645 of the z-values for z0.05. (b) For confidence level 99%, , so Because the area under the standard normal curve to the right of z0.005 is 0.005, the area to the left of z0.005 is 0.9950. We search for the area 0.9950 in the cumulative probability table. The closest entries in the table are 0.9949 and 0.9951, corresponding to z-values 2.57 and 2.58. Because 0.995 is halfway between 0.9949 and 0.9951, we use the average 2.575 of the z-values for z0.005. Example 2 Use the t-distribution table to find the number needed in construction of a confidence interval (page 1, page 2). 1. when the level of confidence is 90% 2. when the level of confidence is 99% Solution: 1. The critical values of t-table gives the value that cuts off a right tail of area for different values of . The last line of that table, the one whose heading is the symbol ∞ for infinity and [z], gives the corresponding -value that cuts off a right tail of the same area . In particular, is the number in that row and in the column with the heading . We read off directly that z=1.645. 2. In the c, is the number in the last row and in the column headed , namely 2.576. The “critical values of t” table can be used to find zc only for those values of c for which there is a column with the heading tc appearing in the table; otherwise, we must use the cumulative probability distribution table in reverse. However, when it can be done, it is both faster and more accurate to use the last line of the table to find zcthan it is to do so using the cumulative probability distribution in reverse. Example 3 A sample of size 49 has sample mean 35 and sample standard deviation 14. Construct a 98% confidence interval for the population mean using this information. Interpret its meaning. Use the critical values of t table (page 1, page 2) for reference. Solution: For confidence level 98%, , so From the t-distribution table, we read directly that Thus, We are 98% confident that the population mean μ lies in the interval [30.3, 39.7], in the sense that, in repeated sampling, 98% of all intervals constructed from the sample data in this manner will contain μ. Example 4 A random sample of 120 students from a large university yields mean GPA 2.71 with sample standard deviation 0.51. Construct a 90% confidence interval for the mean GPA of all students at the university. Solution: For confidence level 90%, , so From the t-distribution table, we read directly that Because n = 120, and s = 0.51, One may be 90% confident that the true average GPA of all students at the university is contained in the interval