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9 SECTION 1 HYPOTHESIS TEST FOR THE DIFFERENCE IN TWO POPULATION PROPORTIONS Two-Population Tests With Qualitative Data  A lot of data are available in the form of proportions or percentages. You see this type of data all the time in the newspaper. Here are some examples taken from the newspaper.  A study of 1049 men and women aged 18 to 65 shows that a greater percentage of women (86%) find it difficult to have sex without emotional involvement compared to men (71 %).

10  A study conducted by an on-line service found that 30% of respondents under age 45 drove sports cars compared to 17% of the 45 or over population.  In each case two populations are being compared and we have taken a sample from each population. What has changed is that the parameter being analyzed is no longer the mean, but the population proportion. SECTION 1 HYPOTHESIS TEST FOR THE DIFFERENCE IN TWO POPULATION PROPORTIONS

11 The Test for Two-Population Proportions  We know that even if the percentage of two populations that have a certain characteristic were exactly the same, we would almost never get exactly the same percentage in two samples from the populations.  This is due to sampling error. The question then becomes, how large a difference in the percentages is large enough to declare the difference statistically significant? SECTION 1 HYPOTHESIS TEST FOR THE DIFFERENCE IN TWO POPULATION PROPORTIONS

12  Our notation will follow the pattern established for the tests of means. SECTION 1 HYPOTHESIS TEST FOR THE DIFFERENCE IN TWO POPULATION PROPORTIONS

13  You can test for a difference in proportions other than zero. The null and alternative hypotheses for the upper-tail and lower-tail tests of differences in proportions are shown below.  Lower-Tail Test Ho:  1   2 Ho:  1 -  2  0 HA:  1 <  2 HA:  1 -  2 < 0 SECTION 1 HYPOTHESIS TEST FOR THE DIFFERENCE IN TWO POPULATION PROPORTIONS

14  Upper-Tail Test H 0 :  1   2 H 0 :  1 -  2  0 H A :  1 >  2 H A :  1 -  2 > 0  Use lower-tail test if you wish to test if the proportion of population 1 is less than the proportion of population 2.  Use upper-tail test if you wish to test if the proportion of population 1 is greater than the proportion of population 2. SECTION 1 HYPOTHESIS TEST FOR THE DIFFERENCE IN TWO POPULATION PROPORTIONS

15 The Test for Two-Population Proportions  The estimate of the true difference in the population proportions  1 -  2, is the difference in the sample proportions, p 1 -p 2.  The standard error of the estimate is similar to the standard error for a single-sample proportion. SECTION 1 HYPOTHESIS TEST FOR THE DIFFERENCE IN TWO POPULATION PROPORTIONS

16  The test statistic is then given by the formula:  The formula for is then: SECTION 1 HYPOTHESIS TEST FOR THE DIFFERENCE IN TWO POPULATION PROPORTIONS

17 SECTION 1 HYPOTHESIS TEST OF THE DIFFERENCE IN TWO POPULATION VARIANCES The F Test for Comparing Population Variances  To decide if we should pool the data we need to test to see if two population variances are equal. Thus, we should use a two-sided test. The null and alternative hypotheses are shown below:  Two-Sided Test

18  Lower-Tail Test  Upper-Tail Test SECTION 1 HYPOTHESIS TEST OF THE DIFFERENCE IN TWO POPULATION VARIANCES

19  Use two-tail test if you wish to test if the variance of population 1 is different from the variance of population 2.  Use lower-tail test if you wish to test if the variance of population 1 is less than the variance of population 2.  Use upper-tail test if you wish to test if the variance of population 1 is greater than the variance of population 2. SECTION 1 HYPOTHESIS TEST OF THE DIFFERENCE IN TWO POPULATION VARIANCES

20  Since we are trying to decide how two population variances compare, it makes sense to compare the sample variances.  Extending this idea to two populations, the point estimate for the ratio of the population variances is the ratio of the sample variances. This is also the test statistic. It is shown below: SECTION 1 HYPOTHESIS TEST OF THE DIFFERENCE IN TWO POPULATION VARIANCES

21  Notice that this ratio is labeled F. This means that the test statistic follows an F-distribution, if the two original populations are normally distributed.  Like the χ ² distribution, which we used to test a single-population variance, the specific shape of the F distribution is determined by its degrees of freedom. But the F distribution has not one, but two values that determine its shape. SECTION 1 HYPOTHESIS TEST OF THE DIFFERENCE IN TWO POPULATION VARIANCES

22  One of these is called the degrees of freedom in the numerator and it is equal to one less than the sample size on which is based, n  The other one is called the degrees of freedom in the denominator and is equal to one less than the sample size on which s ² is based, n SECTION 1 HYPOTHESIS TEST OF THE DIFFERENCE IN TWO POPULATION VARIANCES

23  If you are a doing a two-sided test, then the rejection region is two-sided; if you are doing a one-sided test, then the rejection region is one- sided.  These are shown in Figure SECTION 1 HYPOTHESIS TEST OF THE DIFFERENCE IN TWO POPULATION VARIANCES

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25  The critical values that define the rejection region are labeled F upper, df 1, df 2 and F lower, df 1, df 2.  To find the values for F upper, df 1, df 2 and F lower, df 1, df 2, we need to notice that the shape of the F distribution is not symmetric and the distribution is not centered at zero.  Therefore, the absolute values of F upper and F lower are not the same and they will always be greater than zero. SECTION 1 HYPOTHESIS TEST OF THE DIFFERENCE IN TWO POPULATION VARIANCES

26  In particular, F lower, df 1, df 2 can be found from an upper-tail value as follows:  That is, the lower critical value is found by taking the reciprocal of the upper critical value with the degrees of freedom reversed. Therefore, we need table values only for F upper. SECTION 1 HYPOTHESIS TEST OF THE DIFFERENCE IN TWO POPULATION VARIANCES