Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
SECTION 1 HYPOTHESIS TEST OF THE MEAN: SMALL SAMPLE These steps are summarized here: Step 1: Use the column of the t table that corresponds to the value of you have selected. Step 2: Find the number of degrees of freedom by calculating n-1 and use that row of the t table. Step 3: For upper-tail tests the desired t cutoff value is found at the intersection of that row and column. For lower-tail tests, place a negative sign in front of the value to get the value of tcutoff" 1 SECTION 1 HYPOTHESIS TEST OF THE MEAN: SMALL SAMPLE Degrees of Freedom Degrees of Freedom were first introduced by Fisher. The degrees of freedom of a set of observations are the number of values which could be assigned arbitrarily within the specification of the system. For example, in a sample of size n grouped into k intervals, there are k-1 degrees of freedom, because k-1 frequencies are specified while the other one is specified by the total size n. 2 3 4 SECTION 1 HYPOTHESIS TEST OF THE MEAN: SMALL SAMPLE Summary of Tests of the Mean: There were two major differences from the tests of when is known: a t test statistic was used instead of Z and the cutoff values for the rejection region were found by using the t table instead of the Z table. The rejection regions are summarized next. 5 6 SECTION 1 χ² TEST OF A SINGLE VARIANCE Hypothesis tests of the population variance, ², follow the same basic steps that were used to do a hypothesis test of the population mean and the population proportion. There are two types of situations for which you might be interested in doing a hypothesis test of the population variance. 7 SECTION 1 χ² TEST OF A SINGLE VARIANCE First, you may wish to see if a manufacturing process is running to the specified standard deviation. Second, to perform some other statistical analysis of the data, you may need to know the population variance. In this case you would use sample data to test to see if the population variance is, in fact, a particular value. 8 SECTION 1 χ² TEST OF A SINGLE VARIANCE Two-Tail Hypothesis Test of the Variance To learn how to do a two-sided test of the variance we will complete the five steps of the hypothesis testing procedure for the tissue manufacturer. 9 10 SECTION 1 χ² TEST OF A SINGLE VARIANCE The next step in the hypothesis testing procedure is to select a value for and find the rejection region. To do this we need to know what test statistic will be used in step 3 of the procedure. This is the main difference between a hypothesis test of the population mean, , and the population variance, ². 11 SECTION 1 χ² TEST OF A SINGLE VARIANCE In testing means; we used the sample mean as the basis for our decision to reject or fail to reject the null hypothesis. Because the Central Limit Theorem told us that X-bar has an approximately normal distribution for sufficiently large sample sizes, the appropriate test statistic for a large-sample test of the mean is a Z statistic and thus the rejection region is determined by Z values. 12 SECTION 1 χ² TEST OF A SINGLE VARIANCE If we are testing the population variance then the sample variance, s², will be used as the basis for deciding between H0 and HA. Relying once again on the mathematical statisticians to do the theoretical work, we learn that the sampling distribution associated with the sample variance, s², is called the chi-square (χ²) distribution. 13 SECTION 1 χ² TEST OF A SINGLE VARIANCE The rejection region will be determined by critical values from the chi-square distribution. All example of a chi-square distribution is shown in Figure 12.3. 14 15 SECTION 1 χ² TEST OF A SINGLE VARIANCE The chi-square distribution, just like any distribution, describes how the random variable behaves. If the variable being studied is assumed to be normally distributed, then the statistic to test whether or not the population variance is equal to a particular value is calculated as follows: 16 SECTION 1 χ² TEST OF A SINGLE VARIANCE 2 ( n 1) s 2 2 n = sample size s² = the sample variance ² = the hypothesized value of the population variance under the null hypothesis 17 SECTION 1 χ² TEST OF A SINGLE VARIANCE Like the t distribution, the shape of the chi- square distribution is determined by the number of degrees of freedom. This test statistic has n -1 degrees of freedom. Unlike the Z and t distributions, the χ² distribution is not symmetric. 18 SECTION 1 χ² TEST OF A SINGLE VARIANCE Since this is a two-sided rejection region, the area in each tail must be /2. A typical rejection region is shown as the shaded region in Figure 12.4. The specific values for χ²lower and χ²upper must be found from the chi-square table. A portion of this table is shown in Table 12.1. 19 20 21 22 23 SECTION 1 χ² TEST OF A SINGLE VARIANCE One-Sided Tests of the Variance If you are testing the population variance, most of the time you are interested in doing a twosided test. However, it is possible to do a one-sided test of the variance. The only change in the procedure needed to complete a one-sided test of the variance is in step 2. 24 SECTION 1 χ² TEST OF A SINGLE VARIANCE A one-sided rejection region is used in this case. The rejection regions for the two possibilities are shown in Figure 12.5. Ho: ² [a specific number] HA: ² < [a specific number] Ho: ² [a specific number] HA: ² > [a specific number] 25 26