Inferences for One Population Standard Deviation

Suppose that we want to obtain information about a population standard deviation. If the population is small, we can often determine 𝜎 exactly by first taking a census and then computing 𝜎 from the population data. However, if the population is large, which is usually the case, a census is generally not feasible, and we must use inferential methods to obtain the required information about 𝜎.

Logic Behind

Recall that to perform a hypothesis test with null hypothesis H0: 𝜇 = 𝜇0 for the mean, 𝜇, of a normally distributed variable, we do not use the variable x(bar) as the test statistic; rather, we use the variable t score. Similarly, when performing a hypothesis test with null hypothesis H0: 𝜎 = 𝜎0 for the standard deviatio, 𝜎, of a normally distributed variable, we do not use the variable s as the test statistic; rather, we use a modified version of that variable: This variable has a chi-square distribution. In light of Key Fact 11.2, for a hypothesis test with null hypothesis H0: 𝜎 = 𝜎0, we can use the variable 𝜒2 as the test statistic and obtain the critical value(s) form the 𝜒2-table. We call this hypothesis-testing procedure the one-standard-deviation 𝜒2-test.

Procedure 11.1 gives a step-by-step method for performing a one-standard-deviation 𝜒2-test by using either the critical-value approach or the P-value, but do so is awkward and tedious; thus, we recommend using statistical software.

Unlike the z-tests and t-test for one and two population means, the one-standard-deviation 𝜒2-test is not robust to moderate violations of the normality assumption. In fact, it is so nonrobust that many statisticians advise against its use unless there is considerable evidence that the variable under consideration is normally distributed or very nearly so.

Consequently, before applying Procedure 11.1, construct a normal probability plot. If the plot creates any doubt about the normality of the variable under consideration, do not use Procedure 11.1. We note that nonparametric procedures, which do not require normality, have been developed to perform inferences for a population standard deviation. If you have doubts about the normality of the variable under consideration, you can often use one of those procedures to perform a hypothesis test or find a confidence interval for a population standard deviation. In addition, using Key Fact 11.2, we can also obtain a confidence-interval procedure for a population standard deviation. We call this procedure the one-standard-deviation 𝜒2-interval procedure and present it as Procedure 11.2. This procedure is also known as the 𝜒2-interval procedure for one population standard deviation. This confidence-interval procedure is often formulated in terms of variance instead of standard deviation. Like the one-standard-deviation 𝜒2-test, this procedure is not at all robust to violations of the normality assumption. Inferences for Two Population Standard Deviation, Using Independent Samples

We now introduce hypothesis tests and confidence intervals for two population standard deviations. More precisely, we examine inferences to compare the standard deviations of one variable of two different populations. Such inferences are based on a distribution called the Fdistribution. In many statistical analyses that involve the F-distribution, we also need to determine F-values having areas 0.005, 0.01, 0.025, and 0.10 to their left. Although such F-values aren't available directly from Table VIII, we can obtain them indirectly from the table by using Key Fact 11.4. Logic Behind

To perform hypothesis tests and obtain confidence intervals for two population standard deviations, we need Key Fact 11.5, that is, the distribution of the F-statistic for comparing two population standard deviations. By definition, the F-statistic. In light of Key Fact 11.5, for a hypothesis test with null hypothesis H0: 𝜎1 = 𝜎2 (population standard deviations are equal), we can use the variable F = S12 / S22 as the test statistic and obtain the critical value(s) from the F-table. We call this hypothesis-testing procedure the two-standard-deviations F-test. Procedure 11.3 gives a step-by-step method for performing a two-standard-deviations F-test by using either critical-value approach or the P-value approach.

For the P-value approach, we could use F-table to estimate the P-value, but to do so is awkward and tedious; thus, we recommend using statistical software.

Unlike the z-tests and t-tests for one and two population means, the two-standard-deviation F-test is not robust to moderate violations of the normality assumption. In fact, it is so nonrobust that many statisticians advise against its use unless there is considerable evidence that the variable under consideration is normally distributed, or very nearly so, on each population.

Consequently, before applying Procedure 11.3, construct a normal probability plot of each sample. If either plot creates any doubt about the normality of the variable under consideration, do not use Procedure 11.3. We note that nonparametric procedures, which do not require normality, have been developed to perform inferences for comparing two population standard deviations. If you have doubts about the normality of the variable on the two populations under consideration, you can often use one of those procedures to perform a hypothesis test or find a confidence interval for two population standard deviations.

Using Key Fact 11.5, we can also obtain a confidence-interval procedure, Procedure 11.4, for the ratio of two population standard deviations. We call it the two-standard-deviations F-interval procedure. Also it is known as the F-interval procedure for two population standard deviations and the two-sample F-interval procedure. This confidence-interval procedure is often formulated in terms of variances instead of standard deviations.

To interpret confidence intervals for the ratio 𝜎1 / 𝜎2, of two population standard deviations, considering three cases is helpful.

Case 1: The endpoints of the confidence interval are both greater than 1.

To illustrate, suppose that a 95% confidence interval for 𝜎1 / 𝜎2 is from 5 to 8. Then we can be 95% confident that 𝜎1 / 𝜎2 lies somewhere between 5 and 8 or, equivalently, 5𝜎2 < 𝜎1 < 8𝜎2. Thus, we can be 95% confident that 𝜎1 is somewhere between 5 and 8 times greater than 𝜎2.

Case 2: The endpoints of the confidence interval are both less than 1.

To illustrate, suppose that a 95% confidence interval for 𝜎1 / 𝜎2 is from 0.5 to 0.8. Then we can be 95% confident that 𝜎1 / 𝜎2 lies somewhere between 0.5 and 0.8 or, equivalently, 0.5𝜎2 < 𝜎1 < 0.8𝜎2. Thus, noting that 1/0.5 = 2 and 1/0.8 = 1.25, we can be 95% confident that 𝜎1 < is somewhere between 1.25 and 2 times less than 𝜎2.

Case 3: One endpoint of the confidence interval is less than 1 and the other is greater than 1.

To illustrate, suppose that a 95% confience interval for 5𝜎2 < 𝜎1 < 8𝜎2 is from 0.5 to 8. Then we can be 95% confident that 5𝜎2 < 𝜎1 < 8𝜎2 lies somewhere between 0.5 and 8 or, equivalentluy, 0.5𝜎2 < 𝜎1 < 8𝜎2. Thus, we can be 95% confident that 𝜎1 is somewhere between 2 time less than and 8 times greater than 𝜎2. 