Lecture 42 Section Tue, Apr 8, 2008

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1 the Lecture 42 Section 14.3 Hampden-Sydney College Tue, Apr 8, 2008

2 Outline the 1 2 the 3 4 5

3 the The will compute χ 2 areas, but not χ 2 percentiles. (That s ok.) After performing the χ 2 test by hand, we will see how to do it on the. It turns out that there is no special function for a goodness-of-fit test. We will also see that the χ 2 test with one degree of freedom is equivalent to the single-proportion Z test.

4 - Chi-Square Probabilities the To find a chi-square probability (p-value) on the, Press DISTR. Select χ 2 cdf. Press ENTER. Enter the lower endpoint, the upper endpoint, and the degrees of freedom. Press ENTER. The probability appears.

5 Computing the p-value the In this example, df = 5. To find the p-value, use the to calculate the probability that χ 2 5 would be at least as large as 3.4. p-value = χ 2 cdf(3.4,e99,5) = Accept H 0. We conclude that the die is fair.

6 the Be careful! There is a function called χ 2 -Test, but it does not perform this test. Some TI-84s may have a GOF-Test. To perform a goodness-of-fit test on the, do the following. Put the observed counts in list L 1. Put the hypothetical proportions in list L 1. Multiply L 1 by the sample size and store as L 1. These are the expected counts. Calculate (L 1 -L 2 ) 2 /L 2. Go to LIST > MATH and select sum (item #5). Enter Ans and press ENTER. The value of χ 2 appears. Then use χ 2 cdf to find the p-value.

7 Example - the Suppose we observe 1000 births and find that 520 are male and 480 are female. Does this indicate that male births and female births are not equally likely? Let p 1 = proportion of male births. Let p 2 = proportion of female births. H 0 : p 1 = 0.50, p 2 = 0.50 H 1 : H 0 is not true. α = 0.05.

8 Example - the We have the table Male Female Observed (Expected) (500) (500)

9 Example - the The test statistic is Calculate χ 2 = all cells χ 2 = ( ) = = 1.6 (O E) 2. E ( )2 500

10 Example - the The p-value is p-value = χ 2 cdf(1.6,e99,1) = Accept H 0. The proportion of male births is 50%.

11 Example - the Perform the above test as a two-tailed one-proportion Z test. That is, let the alternative hypothesis be What is the p-value? H 1 : p 1 p 2. What is the value of the test statistic Z? Square that number. What do you get?

12 the The will compute χ 2 areas, but not χ 2 percentiles. (That s ok.) The will not perform the goodness-of-fit test, although we can do it using lists. The goodness-of-fit test with with only two cells is equivalent to the one-proportion Z test.

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