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).

_   X1   X2   X3
Y1   34   43   51
Y2   17   21   17

Answer 1

Solution -

Hypotheses would be-

H0 - There is independence between X and Y values

Ha - There is no independence between X and Y values

we will use contingency table here:-

Table showing Expected frequencies is-

Expected frequency in each cell is calculated using - Row total * Column Total / Grand total

Value of chi-square test statistic is calculated as

= 1.321

P-value will be calculated as P ( > 1.321 ) [Degrees of freedom is calculated as ( 3-1) * (2-1) = 2 ]

= 0.5167 [ Calculated using chi-square tables ]

As p-value is less than 0.01 then null hypothesis is not rejected at 1% level of significance.

Hence we can conclude that there is independence between X and Y values

TY!