Question

The table to the right contains observed values and expected values in parentheses for two categorical​ variables, X and​ Y, where variable X has three categories and variable Y has two categories. Use the table to complete parts ​(a) and ​(b) below. Upper X 1 Upper X 2 Upper X 3 Upper Y 1 34 left parenthesis 35.67 right parenthesis 43 left parenthesis 44.77 right parenthesis 51 left parenthesis 47.56 right parenthesis Upper Y 2 17 left parenthesis 15.33 right parenthesis 21 left parenthesis 19.23 right parenthesis 17 left parenthesis 20.44 right parenthesis ​(a) Compute the value of the​ chi-square test statistic. chi Subscript 0 Superscript 2equals nothing ​(Round to three decimal places as​ needed.) ​(b) Test the hypothesis that X and Y are independent at the alpha equals 0.01 level of significance. A. Upper H 0​: The Upper Y category and Upper X category have equal proportions. Upper H 1​: The proportions are not equal. B. Upper H 0​: The Upper Y category and Upper X category are independent. Upper H 1​: The Upper Y category and Upper X category are dependent. C. Upper H 0​: mu Subscript xequals Upper E Subscript x and mu Subscript yequals Upper E Subscript y Upper H 1​: mu Subscript xnot equals Upper E Subscript x or mu Subscript ynot equals Upper E Subscript y D. Upper H 0​: The Upper Y category and Upper X category are dependent. Upper H 1​: The Upper Y category and Upper X category are independent. What is the​ P-value? ​P-valueequals nothing ​(Round to three decimal places as​ needed.) Should the null hypothesis be​ rejected? A. No​, do not reject Upper H 0. There is not sufficient evidence at the alphaequals0.01 level of significance to conclude that X and Y are dependent because the ​P-valuegreater thanalpha. B. No​, do not reject Upper H 0. There is sufficient evidence at the alphaequals0.01 level of significance to conclude that X and Y are dependent because the ​P-valueless thanalpha. C. Yes​, reject Upper H 0. There is not sufficient evidence at the alphaequals0.01 level of significance to conclude that X and Y are dependent because the ​P-valuegreater thanalpha. D. Yes​, reject Upper H 0. There is not sufficient evidence at the alphaequals0.01 level of significance to conclude that X and Y are dependent because the ​P-valueless thanalpha. Click to select your answer(s).

Answer 1

Steps followed for Chi Square Test

The following cross tablulation have been provided. The row and column total have been calculated and they are shown below: Column1 34 17 51 Column 2 43 21 64 Column 3 51 17 68 Total Row 1 Row 2 Total 128 183 The expected values are computed in terms of row and column totals. In fact, the formula is E- HC, 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 Row 1 Row 2 Total Column1 부 = 35.672 16515.328 51 Column 2 Column 3 Total = 44.765 =19.235 64 = 47.563 .437 68 128 15.328 183 Based on the observed and expected values, the squared distances can be computed according to the following formula: (E O2 E. The table with squared distances is shown below: Squared Distances Row 1 Row 2 Column1 Column 2 43-44.7650.07 21-19.235 0.162 Column 3 1 475630.248 し34735 67212 0.078 17-15.328 0.182 -= 0.578

the The following null and alternative hypotheses need to be tested: Ho: The two variables are independent H The two variables are dependent This corresponds to a Chi-Square test of independence. Based on the information provided, the significance level is α 0.01 , the number of degrees of freedom is df = (2-1) × (8-1) = 2, so then the rejection region for this test is R = -"X2 > 9.21) Test Statistics The Chi-Squared statistic is computed as follows: 20.078+0.182+0.07 +0.162+0.248 +0.578 1.319 %2+ Since it is observed that х"-1319 < 9.21, it is then concluded that the null hypothesis is not rejected Conclusion It is concluded that the null hypothesis Ho is not rejected. Therefore, there is NOT enough evidence to claim that the two variables are dependent, at the 0.01 significance level.

From the above steps, we can answer all parts:

Chi Square Statistics = 1.319 (Check calculation above)

The following null and alternative hypotheses need to be tested (Option B are correct):

H0​: X and Y are independent

Ha​: X and Y are dependent

P-value can be obtained using online calculator for df = (2-1)*(3-1) = 2 and Chi-square statistics = 1.319 (Screenshot attached)

P-value = 0.517

P Value from Chi-square Calculator This calculator is designed to generate a p-value from a chi square score. If you need to derive a chi-square score from raw data, you should use our chi-square calculator (which will additionally calculate the p-value for you). The calculator below should be self-explanatory, but just in case it's not: your chi-square score goes in the chi-square score box, you stick your degrees of freedom in the DFbox (df= (NColumns-7)*(NRows-1) for chi-square test for button. 1.319 Chi-square score: DF Significance Level: ) 0.01 ● 005 0.10 The P-Value is .51711. The result is not significant at p <.01

Since p-value (0.517) is greater than 0.01 level of significance, we fail to reject null hypothesis. There is not sufficient evidence at the alpha equals 0.01 level of significance to conclude that X and Y are dependent because the ​P-value greater than alpha (Option A are correct).