The GoodnessofFit Test


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1 The GoodnessofFit Test Lecture 49 Section 14.3 Robb T. Koether HampdenSydney College Tue, Apr 24, 2012 Robb T. Koether (HampdenSydney College) The GoodnessofFit Test Tue, Apr 24, / 15
2 Outline 1 GoodnessofFit Test on the TI83 2 The Effect of Sample Size 3 GoodnessofFit Example 4 Assignment Robb T. Koether (HampdenSydney College) The GoodnessofFit Test Tue, Apr 24, / 15
3 Outline 1 GoodnessofFit Test on the TI83 2 The Effect of Sample Size 3 GoodnessofFit Example 4 Assignment Robb T. Koether (HampdenSydney College) The GoodnessofFit Test Tue, Apr 24, / 15
4 GoodnessofFit Test on the TI83 Be careful when using the TI83! There is a function called χ 2 Test, but it does not perform the goodnessoffit test. Some TI84s have a GOFTest function. The GOFTest function does perform the goodnessoffit test. Robb T. Koether (HampdenSydney College) The GoodnessofFit Test Tue, Apr 24, / 15
5 GoodnessofFit Test on the TI83 TI83 Goodnessoffit test Put the observed counts in list L 1. Put the hypothetical proportions in list L 2. Multiply L 2 by the sample size and store as L 2. These are the expected counts. Calculate (L 1 L 2 ) 2 /L 2 (either all at once or step by step). 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 pvalue. Robb T. Koether (HampdenSydney College) The GoodnessofFit Test Tue, Apr 24, / 15
6 Outline 1 GoodnessofFit Test on the TI83 2 The Effect of Sample Size 3 GoodnessofFit Example 4 Assignment Robb T. Koether (HampdenSydney College) The GoodnessofFit Test Tue, Apr 24, / 15
7 The Effect of Sample Size In the previous example, 3 s and 5 s were overrepresented and 1 s were underrepresented. Robb T. Koether (HampdenSydney College) The GoodnessofFit Test Tue, Apr 24, / 15
8 The Effect of Sample Size In the previous example, 3 s and 5 s were overrepresented and 1 s were underrepresented. Yet that was not significant. Robb T. Koether (HampdenSydney College) The GoodnessofFit Test Tue, Apr 24, / 15
9 The Effect of Sample Size In the previous example, 3 s and 5 s were overrepresented and 1 s were underrepresented. Yet that was not significant. What if we observed the same proportions, but for a larger sample? Robb T. Koether (HampdenSydney College) The GoodnessofFit Test Tue, Apr 24, / 15
10 The Effect of Sample Size In the previous example, 3 s and 5 s were overrepresented and 1 s were underrepresented. Yet that was not significant. What if we observed the same proportions, but for a larger sample? That is, what if we had observed Number Observed (Expected) (150) (150) (150) (150) (150) (150) Robb T. Koether (HampdenSydney College) The GoodnessofFit Test Tue, Apr 24, / 15
11 The Effect of Sample Size In the previous example, 3 s and 5 s were overrepresented and 1 s were underrepresented. Yet that was not significant. What if we observed the same proportions, but for a larger sample? That is, what if we had observed Number Observed (Expected) (150) (150) (150) (150) (150) (150) Would it be signficant? Robb T. Koether (HampdenSydney College) The GoodnessofFit Test Tue, Apr 24, / 15
12 Outline 1 GoodnessofFit Test on the TI83 2 The Effect of Sample Size 3 GoodnessofFit Example 4 Assignment Robb T. Koether (HampdenSydney College) The GoodnessofFit Test Tue, Apr 24, / 15
13 Example  Fairness of a Coin Example (TI83 Goodnessoffit test) Suppose we toss a coin 1000 times and get 525 heads and 475 tails. Does this indicate that is fair or that it is biased? Robb T. Koether (HampdenSydney College) The GoodnessofFit Test Tue, Apr 24, / 15
14 Example  Fairness of a Coins Example (TI83 Goodnessoffit test) (1) Let p 1 = proportion of heads. Let p 2 = proportion of tails. H 0 : p 1 = 0.50, p 2 = 0.50 H 1 : H 0 is not true. (2) α = (3) The test statistic is χ 2 = all cells (O E) 2. E Robb T. Koether (HampdenSydney College) The GoodnessofFit Test Tue, Apr 24, / 15
15 Example  Fairness of a Coin Example (TI83 Goodnessoffit test) (4) We have the table Calculate Heads Tails Observed (Expected) (500) (500) χ 2 = = = = 2.5. ( ) ( )2 500 Robb T. Koether (HampdenSydney College) The GoodnessofFit Test Tue, Apr 24, / 15
16 Example  Fairness of a Coin Example (TI83 Goodnessoffit test) (5) The pvalue is pvalue = χ 2 cdf(2.5,e99,1) = (6) Accept H 0. (7) The proportion of heads is 50%. Robb T. Koether (HampdenSydney College) The GoodnessofFit Test Tue, Apr 24, / 15
17 Example  Fairness of a Coin Perform the above test as a twotailed oneproportion z test. That is, let the alternative hypothesis be H 1 : p 1 p 2. What is the pvalue? What is the value of the test statistic z? Square that number. What do you get? Robb T. Koether (HampdenSydney College) The GoodnessofFit Test Tue, Apr 24, / 15
18 Outline 1 GoodnessofFit Test on the TI83 2 The Effect of Sample Size 3 GoodnessofFit Example 4 Assignment Robb T. Koether (HampdenSydney College) The GoodnessofFit Test Tue, Apr 24, / 15
19 Assignment Homework Read Sections , pages Let s Do It! 14.2, Exercises 611, 14, 15, page 935. Robb T. Koether (HampdenSydney College) The GoodnessofFit Test Tue, Apr 24, / 15
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