Section 8.2 Confidence Intervals for a Population Mean, Standard Deviation Unknown 367 discovery to be proprietary information, and forbade Gosset to publish it. He published it anyway, using the pseudonym ‘‘Student.’’ Gosset had done this before; see Section 6.3. Recall: Degrees of freedom were introduced in Section 3.2. In fact, there are many different Student’s t distributions; they are distinguished by a quantity called the degrees of freedom. When using the Student’s t distribution to construct a confidence interval for a population mean, the number of degrees of freedom is 1 less than the sample size. Degrees of Freedom for the Student’s t Distribution When constructing a confidence interval for a population mean, the number of degrees of freedom for the Student’s t distribution is 1 less than the sample size n. number of degrees of freedom = n − 1 Figure 8.9 presents t distributions for several different degrees of freedom, along with a standard normal distribution for comparison. The t distributions are symmetric and unimodal, just like the normal distribution. The t distribution is more spread out than the standard normal distribution, because the sample standard deviation s is, on the average, a bit less than ��. When the number of degrees of freedom is small, this tendency is more pronounced, so the t distributions are much more spread out than the normal. When the number of degrees of freedom is large, s tends to be very close to ��, so the t distribution is very close to the normal. Figure 8.9 shows that with 10 degrees of freedom, the difference between the t distribution and the normal is not great. If a t distribution with 30 degrees of freedom were plotted in Figure 8.9, it would be indistinguishable from the normal distribution. 0 t with 10 degrees of freedom Standard normal t with 1 degree of freedom t with 4 degrees of freedom Figure 8.9 Plots of the Student’s t distribution for 1, 4, and 10 degrees of freedom. The standard normal distribution is plotted for comparison. The t distributions are more spread out than the normal, but the amount of extra spread decreases as the number of degrees of freedom increases. SUMMARY The Student’s t distribution has the following properties: ∙ It is symmetric and unimodal. ∙ It is more spread out than the standard normal distribution. ∙ If we increase the number of degrees of freedom, the Student’s t curve becomes closer to the standard normal curve. Area = 1 − α Area = α/2 Area = α/2 −tα/2 0 tα/2 Figure 8.10 Finding the critical value We use the Student’s t distribution to construct confidence intervals for �� when �� is unknown. The idea behind the critical value is the same as for the normal distribution. To find the critical value for a confidence interval with a given level, let 1 − �� be the confidence level expressed as a decimal. The critical value is then t��∕2, because the area under the Student’s t curve between −t��∕2 and t��∕2 is 1 − ��. See Figure 8.10. The critical value t��∕2 can be found in Table A.3, in the row corresponding to the number of degrees of freedom and the column corresponding to the desired confidence level. Example 8.10 shows how to find a critical value.
navidi_monk_elementary_statistics_2e_ch7-9
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