Question

Perform a chi-square independence test using the critical value approach, provided the conditions for using the test are met. Be sure to state the hypotheses and the significance level, to obtain the expected frequencies, to obtain the critical value, to compute the value of the test statistic, and to state your conclusion. 160 students who were majoring in either math or English were asked a test question, and the researcher recorded whether they answered the question correctly. The sample results are given below. At the 0.10 significance level, do the data provide sufficient evidence to support the claim that an association exists between response and major- Response and Major are dependent? Test for Independence-Contingency Tables-Chi-Square One Right Tail Test Correct Math 27 English Incorrect 53 37

Answer 1

Solution

The following cross tablulation have been provided. The row and column total have been calculated and they are shown below: Correct Incorrect Total Math 27 53 80 English 43 37 80 Total 70 90 160 The expected values are computed in terms of row and column totals. In fact, the formula is Eij = RxC, where R; corresponds to the total sum of elements in row i, C; corresponds to the total sum of elements in column j, and T is the grand total. The table below shows the calculations to obtain the table with expected values: Expected Values Correct Incorrect Total Math 70X80 160 35 90 x 80 160 - 45 80 English 70 x 80 160 = 35 90X80 160 = = 45 80 Total 70 90 160 Based on the observed and expected values, the squared distances can be computed according to the following formula: (E - 0)2/E. The table with squared distances is shown below:

Squared Distances Correct Incorrect Math (27–352 35 1.829 (53-45) 45 1.422 English (43–35) 35 1.829 (37-45) = 1.422 45 Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: Ho: The two variables are independent He: The two variables are dependent This corresponds to a Chi-Square test of independence. Rejection Region Based on the information provided, the significance level is a = 0.10, the number of degrees of freedom is df = (2 – 1) (2 - 1) = 1, so then the rejection region for this test is R= {x? : X?> 2.706} Test Statistics The Chi-Squared statistic is computed as follows: n x = Σ = [ (Oij – E;;) = = 1.829 +1.829 +1.422 +1.422 = = 6.502 Ej i=1

Decision about the null hypothesis = Since it is observed that x rejected. = 6.502 > xả = 2.706, it is then concluded that the null hypothesis is

Conclusion : We have enough evidence to claim that association exists between response and major-Response and Major are dependent.