Lecture 45 Sections Wed, Nov 19, 2008

Transcription

1 The Lecture 45 Sections 14.5 Hampden-Sydney College Wed, Nov 19, 2008

2 Outline The The 4 5

3 The Exercise 14.20, page 949. A certain job in a car assembly plant involves a great deal of stress. A study was conducted to assess whether there is any difference in how men versus women adjust to the stress. A random sample of 25 men and 25 women employed in this job were surveyed. The results are given as follows:

4 The Exercise 14.20, page 949. Well Adjusted Not Well Adjusted Men 18 7 Women At the 1% significance level, test whether the adjustment status is the same for men versus women.

5 The Solution (1) H 0 : The populations are homogeneous. H 1 : The populations are not homogeneous. (2) α = (3) χ 2 = all cells (O E) 2. E

6 The Solution [ ] 18 7 (4) Enter the matrix as matrix A in the TI Use STAT > TEST > χ 2 -Test. The calculator reports that χ 2 = (5) The calculator also reports that p-value = (6) Accept H 0. (7) The adjustment statuses of men and women are the same.

7 The Why are degrees of freedom called degrees of freedom?

8 The Consider a 4 6 table of observed values. Suppose that we are told only the row and column totals. A B C D E F Row Total Col Total

9 The How many values can we fill in in Row 1 arbitrarily and still make the total equal to 40? A B C D E F Row Total 1?????? Col Total

10 We can fill in all but one value. The A B C D E F Row Total ? Col Total

11 The The last value is forced to be 2 in order to make the total equal 40. A B C D E F Row Total Col Total

12 The same is true for any of the rows. The A B C D E F Row Total 1????? x 40 2????? x 60 3????? x 50 4????? x 90 Col Total

13 The How many rows can be filled in that way before the remaining rows are forced? A B C D E F Row Total 1????? x 40 2????? x 60 3????? x 50 4????? x 90 Col Total

14 All but the last row can be filled in arbitrarily. The A B C D E F Row Total x x x x x x 90 Col Total

15 The last row is forced. The A B C D E F Row Total Col Total

16 The Therefore, the number of degrees of freedom is (No. of rows 1) (No. of cols 1). A B C D E F Row Total Col Total

17 Independent Variables The Definition (Independent variables) Two variables are independent if each one has nothing to do with the other. That is, knowing the value of either variable is of no benefit in predicting the value of the other variable. Whether it rains is independent of the day of the week. However, whether it rains is not independent of the month of the year.

18 The The The test of independence is identical to the test of homogeneity.

19 The The Example ( ) On the TI-83, when we use randint(1,100), we expect to get even integers half the time and odd integers half the time. Does it depend on whether the seed is even or odd? That is, is the even/odd-ness of the random number independent of the even/odd-ness of the seed?

20 The The Example ( ) I experimented with 30 even seeds and 30 odd seeds. The seeds themselves were selected at random. In each case, I noted whether the seed was even and whether randint(1,100) was even.

21 The The Example ( ) I obtained the following data. randint(1,100) Even Odd Seed Even Odd 8 22

22 The The Example ( ) Find the expected counts. Then run the test of independence at the 1% level of significance.

23 The The Example ( ) (1) H 0 : The variables are independent. H 1 : The variables are not independent. (2) α = (3) χ 2 = all cells (O E) 2. E

24 The The Example ( ) (4) The value of the chi-square statistic is χ 2 = (20 14) (22 16) 2 16 = (10 16) (8 14) (5) The p-value is χ 2 cdf(9.643,e99,1) = (6) Reject H 0. (7) The parity of the seed and the parity of the random integer are not independent!

25 The Read Section 14.5, pages Let s Do It! 14.6, (14.6 is an example of Simpson s paradox.) Exercises 23-27, 31, 34, page 959. (Ex. 31 and 34 are examples of Simpson s paradox.) Chapter 35-40, 42, 47, page 966.