the table below summarizes data from a survey of a sample of women. using a 0.01 significance level, and…

the table below summarizes data from a survey of a sample of women. using a 0.01 significance level, and assuming that the sample sizes of 900 men and 400 women are predetermined, test the claim that the proportions of agree/disagree responses are the same for subjects interviewed by men and the subjects interviewed by women. does it appear that the gender of the interviewer affected the responses of women?\n\n| gender of interviewer | man | woman |\n|--|--|--|\n| women who agree | 588 | 351 |\n| women who disagree | 312 | 49 |\n\nclick here to view the chi - square distribution table.\n\nidentify the null and alternative hypotheses. choose the correct answer below.\na. $h_0$: the proportions of agree/disagree responses are the same for the subjects interviewed by men and the subjects interviewed by women.\n$h_1$: the proportions are different.\nb. $h_0$: the response of the subject and the gender of the subject are independent.\n$h_1$: the response of the subject and the gender of the subject are dependent.\nc. $h_0$: the proportions of agree/disagree responses are different for the subjects interviewed by men and the subjects interviewed by women.\n$h_1$: the proportions are the same.\n\ncompute the test statistic.\n(round to three decimal places as needed.)

the table below summarizes data from a survey of a sample of women. using a 0.01 significance level, and assuming that the sample sizes of 900 men and 400 women are predetermined, test the claim that the proportions of agree/disagree responses are the same for subjects interviewed by men and the subjects interviewed by women. does it appear that the gender of the interviewer affected the responses of women?\n\n| gender of interviewer | man | woman |\n|--|--|--|\n| women who agree | 588 | 351 |\n| women who disagree | 312 | 49 |\n\nclick here to view the chi - square distribution table.\n\nidentify the null and alternative hypotheses. choose the correct answer below.\na. $h_0$: the proportions of agree/disagree responses are the same for the subjects interviewed by men and the subjects interviewed by women.\n$h_1$: the proportions are different.\nb. $h_0$: the response of the subject and the gender of the subject are independent.\n$h_1$: the response of the subject and the gender of the subject are dependent.\nc. $h_0$: the proportions of agree/disagree responses are different for the subjects interviewed by men and the subjects interviewed by women.\n$h_1$: the proportions are the same.\n\ncompute the test statistic.\n(round to three decimal places as needed.)

Answer

Explanation:

Step1: Calculate row - totals and column - totals

Row 1 (agree) total: $588 + 351=939$ Row 2 (disagree) total: $312 + 49 = 361$ Column 1 (man) total: $588+312 = 900$ Column 2 (woman) total: $351 + 49=400$ Grand total: $939+361=1300$

Step2: Calculate expected frequencies

The formula for the expected frequency $E_{ij}=\frac{R_i\times C_j}{n}$, where $R_i$ is the $i$-th row total, $C_j$ is the $j$-th column total, and $n$ is the grand total. For the cell of women who agree and interviewed by man: $E_{11}=\frac{939\times900}{1300}\approx646.154$ For the cell of women who agree and interviewed by woman: $E_{12}=\frac{939\times400}{1300}\approx292.846$ For the cell of women who disagree and interviewed by man: $E_{21}=\frac{361\times900}{1300}\approx253.846$ For the cell of women who disagree and interviewed by woman: $E_{22}=\frac{361\times400}{1300}\approx107.154$

Step3: Calculate the chi - square test statistic

The formula for the chi - square test statistic is $\chi^2=\sum\frac{(O - E)^2}{E}$, where $O$ is the observed frequency and $E$ is the expected frequency. For the first cell: $\frac{(588 - 646.154)^2}{646.154}\approx5.237$ For the second cell: $\frac{(351-292.846)^2}{292.846}\approx11.599$ For the third cell: $\frac{(312 - 253.846)^2}{253.846}\approx13.397$ For the fourth cell: $\frac{(49 - 107.154)^2}{107.154}\approx31.767$ $\chi^2=5.237 + 11.599+13.397+31.767\approx61.999$

Answer:

$62.000$