EXAMPLE 5 p-Value method for the goodness of fit test using technology
Table 5 contains the distribution of violent crime in New York City in 2012.2 Suppose that a random sample of 1000 violent crimes in New York City yielded the counts shown in Table 6. Test whether the population proportions have changed since 2012, using the p-value method and level of significance .
Murder | Rape | Robbery | Assault |
---|---|---|---|
0.01 | 0.04 | 0.35 | 0.60 |
640
Murder | Rape | Robbery | Assault |
---|---|---|---|
6 | 50 | 350 | 594 |
Solution
Reject if the p-value .
What Results Might We Expect?
Before we do the formal hypothesis test, let's try to figure out what the conclusion might be. Figure 4 is a clustered bar graph (see Section 2.1) of the observed and expected frequencies for each of the four categories. If were true, then, for each category, we would expect the red bars (observed frequencies) and blue bars (expected frequencies) to have somewhat similar heights. In fact, the heights of the bars are fairly similar for all four categories, indicating not much difference between the crimes that were observed and the crimes that were expected. Thus, we might expect to not reject .
First, we need to find the expected frequencies. We have , so the expected frequencies are as shown here.
Category | |
---|---|
Murder | |
Rape | |
Robbery | |
Assault |
641
Next, check the conditions for this test. Because (a) none of the expected frequencies is less than 1 and (b) no more than 20% of the expected frequencies are less than 5, we may proceed. We use the instructions provided in the Step-by-Step Technology Guide at the end of this section.
Step 2 Find the test statistic . The TI-83/84 results in Figure 5 tell us that
Step 3 Find the p-value. Figure 5 also tells us that
This p-value, for the χ2 distribution with 3 degrees of freedom, is shown in Figure 6.
Figure 7a shows the TI-84 output for the test, and Figure 7b shows the SPSS output for the test, confirming our test statistic of 4.16 and p-value of 0.2447.
NOW YOU CAN DO
Exercises 23–26.