Equivalent Tests A x^2 test involving a 2 x 2 table is equival...

Question

Equivalent Tests A x^2 test involving a 2 x 2 table is equivalent to the test for the difference between two proportions, as described in Section 9-1. Using Table 11-1 from the Chapter Problem, verify that the x^2 test statistic and the z test statistic (found from the test of equality of two proportions) are related as follows: z^2 = x^2 Also show that the critical values have that same relationship.

Accepted Answer

A school conducted an experiment to compare the proportion of students who passed a short quiz after using two different study methods. The results are summarized in the 2x2 table below. They were given a 2 by 2 table with different methods and whether the student passed or not. Using this data, conduct a chi square test for independence using a 2x2 contingency table and a 2 proportion Z test to compare the two groups. Then show how the two test statistics are related by verifying that Z2 equals 2. Also, demonstrate that the critical values from both tests have the same squared relationship. Use a 0.05 significance level for both tests. Now, to solve this, let's find our critical values and our test statistics. Now, to find our chi square test statistic, we'll have to first find our expected values, because we have our observed value table. Now our expected values will first add together the sum of each column and each row. For Me A, we have 50. Method B, we also have 50. For past we have 70 and did not pass, we have 30. And then our grand total will be 100. Now let's calculate our expected values. We will multiply the row 50. By the column 70 and divided by the total. For our next part of the table, we will multiply 50 by its columns some 30, divided by 100. Then we have 50 Multiplied. By 70. Divided by 100 And 50, multiplied by 30, divided by 100. This gives us a new expected value table. We will then have 50 multiplied by 70, 3500 divided by 100. It is 35 50 multiplied by 30, divided by 100, gives us 15. We now have our expected value table. Now let's find our chi squared statistic. That is given by the sun. Of the difference of our observed minus our expected value squared. Divided by the expected value. Which in our case will have 30. Minus 35 squared divided by 30. 5 1st. Plus 20. Minus 15 squared. Divided by 15 Plus 40 minus 35 squared. Divided by 35. Plus 15 minus 10 square divided by 15. And so we can calculate this and sum all of our values together, but we should get 4.76 as archi square statistic. Now, let's find our Z statistic. Since we're using proportions, we'll first find proportion one. Which in our case 30 divided by 50. Will give us a 0.6. This is the passing rate for method 8. Will the same for method be? Being 40 divided by 50. Which gives us 0.8. We can now find our overall pole proportion, which is given by PHAT. Equals 30 plus 40. Divided by 100 Which in our case, is 0.7. We can now find our standard error. Which is given by the square root. Of Phat multiplied by 1 minus Phat, which is all multiplied. By 1 divided by N1 plus 1 divided by N2. We have the square root then. Of 0.7, multiplied by 0.3. Multiplied by 1 divided by 50 plus 1 divided by 50. Which gives us approximately 0.0916. This means that our Z. will be 0.6. Minus 0.8 for difference in proportions, divided by the standard error. 0. 0916, which is approximately -2.185. And the squared, if we square that value, will be approximately 4.77. We can then see that our chi squared. Being 4.76 is approximately the same as our Z squared being 4.77. So we can confirm that these two are equals to each other. Now, we also need to check if the critical values are the same value. So for the critical value, The Z critical value For 0.05 is the alpha is 1.96. That means that the critical squared. is approximately 3.84. For the chi square critical. We will look at the chi square table. At 0.05 with the degrees of freedom of 1. This will give us 3.84. We can then see that the critical values are also equal to each other. And based on what we found and what we calculated, We can then confirm That the equivalent for the 2 by 2 table and the 2 proportion is true. OK, I hope to help you solve the problem. Thank you for watching. Goodbye.

Key Concepts

Chi-Squared Test

Z-Test for Proportions

Relationship Between Chi-Squared and Z-Test

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