Confidence Intervals for the Mean (s Unknown) 6.2 310 CHAPTER 6 Confidence Intervals What You Should Learn How to interpret the t-distribution and use a t-distribution table How to construct and interpret confidence intervals for a population mean when s is not known The t@Distribution Confidence Intervals and t@Distributions The t-Distribution In many real-life situations, the population standard deviation is unknown. So, how can you construct a confidence interval for a population mean when s is not known? For a simple random sample that is drawn from a population that is normally distributed or has a sample size of 30 or more, you can use the sample standard deviation s to estimate the population standard deviation s. However, when using s, the sampling distribution of x does not follow a normal distribution. In this case, the sampling distribution of x follows a t-distribution. If the distribution of a random variable x is approximately normal, then t = x - m s 1n follows a t@distribution. Critical values of t are denoted by tc. Here are several properties of the t@distribution. 1. The mean, median, and mode of the t@distribution are equal to 0. 2. The t@distribution is bell-shaped and symmetric about the mean. 3. The total area under the t@distribution curve is equal to 1. 4. The tails in the t@distribution are “thicker” than those in the standard normal distribution. 5. The standard deviation of the t@distribution varies with the sample size, but it is greater than 1. 6. The t@distribution is a family of curves, each determined by a parameter called the degrees of freedom. The degrees of freedom (sometimes abbreviated as d.f.) are the number of free choices left after a sample statistic such as x is calculated. When you use a t@distribution to estimate a population mean, the degrees of freedom are equal to one less than the sample size. d.f. = n - 1 Degrees of freedom 7. As the degrees of freedom increase, the t@distribution approaches the standard normal distribution, as shown in the figure. For 30 or more degrees of freedom, the t@distribution is close to the standard normal distribution. d.f. = 2 d.f. = 5 Standard normal curve 0 t DEFINITION Study Tip Here is an example that illustrates the concept of degrees of freedom. The number of chairs in a classroom equals the number of students: 25 chairs and 25 students. Each of the first 24 students to enter the classroom has a choice on which chair he or she will sit. There is no freedom of choice, however, for the 25th student who enters the room.
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