Chi-Square vs. z Section 25.9

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1 Chi-Square vs. z Section 25.9 Lecture 49 Robb T. Koether Hampden-Sydney College Wed, Apr 20, 2016 Robb T. Koether (Hampden-Sydney College) Chi-Square vs. zsection 25.9 Wed, Apr 20, / 11

2 Outline 1 χ 2 Versus z Goodness-of-fit Test, 2 Categories 2 2 Table 2 Assignment Robb T. Koether (Hampden-Sydney College) Chi-Square vs. zsection 25.9 Wed, Apr 20, / 11

3 Example There are two situations where the χ 2 test has 1 degree of freedom. A goodness-of-fit test with only 2 categories. A two-way table with 2 rows and 2 columns. Robb T. Koether (Hampden-Sydney College) Chi-Square vs. zsection 25.9 Wed, Apr 20, / 11

4 Example There are two situations where the χ 2 test has 1 degree of freedom. A goodness-of-fit test with only 2 categories. A two-way table with 2 rows and 2 columns. In these cases, the test could be performed as a z-test. Goodness-of-fit test Test of one proportion. 2 2 table Comparing two proportions. Robb T. Koether (Hampden-Sydney College) Chi-Square vs. zsection 25.9 Wed, Apr 20, / 11

5 Outline 1 χ 2 Versus z Goodness-of-fit Test, 2 Categories 2 2 Table 2 Assignment Robb T. Koether (Hampden-Sydney College) Chi-Square vs. zsection 25.9 Wed, Apr 20, / 11

6 Goodness-of-fit Test, 2 Categories To test a coin for fairness, we toss it 1000 times and get 467 heads and 533 tails. Robb T. Koether (Hampden-Sydney College) Chi-Square vs. zsection 25.9 Wed, Apr 20, / 11

7 Goodness-of-fit Test, 2 Categories To test a coin for fairness, we toss it 1000 times and get 467 heads and 533 tails. Using the z-test: Robb T. Koether (Hampden-Sydney College) Chi-Square vs. zsection 25.9 Wed, Apr 20, / 11

8 Goodness-of-fit Test, 2 Categories To test a coin for fairness, we toss it 1000 times and get 467 heads and 533 tails. Using the z-test: The sample proportion is ˆp = Robb T. Koether (Hampden-Sydney College) Chi-Square vs. zsection 25.9 Wed, Apr 20, / 11

9 Goodness-of-fit Test, 2 Categories To test a coin for fairness, we toss it 1000 times and get 467 heads and 533 tails. Using the z-test: The sample proportion is ˆp = The test statistic is z = (0.5)(0.5) 1000 = = Robb T. Koether (Hampden-Sydney College) Chi-Square vs. zsection 25.9 Wed, Apr 20, / 11

10 Goodness-of-fit Test, 2 Categories Using the goodness-of-fit test: Robb T. Koether (Hampden-Sydney College) Chi-Square vs. zsection 25.9 Wed, Apr 20, / 11

11 Goodness-of-fit Test, 2 Categories Using the goodness-of-fit test: The observed and expected counts: Heads Tails Total Obs Exp Robb T. Koether (Hampden-Sydney College) Chi-Square vs. zsection 25.9 Wed, Apr 20, / 11

12 Goodness-of-fit Test, 2 Categories Using the goodness-of-fit test: The observed and expected counts: The test statistic is Heads Tails Total Obs Exp χ 2 = ( ) ( )2 500 = = Robb T. Koether (Hampden-Sydney College) Chi-Square vs. zsection 25.9 Wed, Apr 20, / 11

13 Outline 1 χ 2 Versus z Goodness-of-fit Test, 2 Categories 2 2 Table 2 Assignment Robb T. Koether (Hampden-Sydney College) Chi-Square vs. zsection 25.9 Wed, Apr 20, / 11

14 2 2 Table To test two coins to see whether they land heads equally often, we toss each coin 1000 times. Robb T. Koether (Hampden-Sydney College) Chi-Square vs. zsection 25.9 Wed, Apr 20, / 11

15 2 2 Table To test two coins to see whether they land heads equally often, we toss each coin 1000 times. The first coin lands heads 600 times; the second coin lands heads 550 times. Robb T. Koether (Hampden-Sydney College) Chi-Square vs. zsection 25.9 Wed, Apr 20, / 11

16 2 2 Table To test two coins to see whether they land heads equally often, we toss each coin 1000 times. The first coin lands heads 600 times; the second coin lands heads 550 times. Using the z-test: Robb T. Koether (Hampden-Sydney College) Chi-Square vs. zsection 25.9 Wed, Apr 20, / 11

17 2 2 Table To test two coins to see whether they land heads equally often, we toss each coin 1000 times. The first coin lands heads 600 times; the second coin lands heads 550 times. Using the z-test: The sample proportions are ˆp 1 = and ˆp 2 = Robb T. Koether (Hampden-Sydney College) Chi-Square vs. zsection 25.9 Wed, Apr 20, / 11

18 2 2 Table To test two coins to see whether they land heads equally often, we toss each coin 1000 times. The first coin lands heads 600 times; the second coin lands heads 550 times. Using the z-test: The sample proportions are ˆp 1 = and ˆp 2 = The pooled estimate is ˆp = = = Robb T. Koether (Hampden-Sydney College) Chi-Square vs. zsection 25.9 Wed, Apr 20, / 11

19 2 2 Table To test two coins to see whether they land heads equally often, we toss each coin 1000 times. The first coin lands heads 600 times; the second coin lands heads 550 times. Using the z-test: The sample proportions are ˆp 1 = and ˆp 2 = The pooled estimate is ˆp = = = The test statistic is z = (0.575)( ) ( ) = = Robb T. Koether (Hampden-Sydney College) Chi-Square vs. zsection 25.9 Wed, Apr 20, / 11

20 2 2 Table Using the goodness-of-fit test: Robb T. Koether (Hampden-Sydney College) Chi-Square vs. zsection 25.9 Wed, Apr 20, / 11

21 2 2 Table Using the goodness-of-fit test: The 2 2 table is Heads Tails Total Coin Coin Robb T. Koether (Hampden-Sydney College) Chi-Square vs. zsection 25.9 Wed, Apr 20, / 11

22 2 2 Table Using the goodness-of-fit test: The 2 2 table is Heads Tails Total Coin Coin Robb T. Koether (Hampden-Sydney College) Chi-Square vs. zsection 25.9 Wed, Apr 20, / 11

23 2 2 Table Using the goodness-of-fit test: The 2 2 table is The test statistic is Heads Tails Total Coin Coin χ 2 = ( ) ( ) ( ) ( )2 425 = = Robb T. Koether (Hampden-Sydney College) Chi-Square vs. zsection 25.9 Wed, Apr 20, / 11

24 Outline 1 χ 2 Versus z Goodness-of-fit Test, 2 Categories 2 2 Table 2 Assignment Robb T. Koether (Hampden-Sydney College) Chi-Square vs. zsection 25.9 Wed, Apr 20, / 11

25 Assignment Assignment Read Sections Robb T. Koether (Hampden-Sydney College) Chi-Square vs. zsection 25.9 Wed, Apr 20, / 11

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