Q: Which of the following is the correct formula for the expected frequency $$E_{r,c} in a $$contingency table cell, assuming independence?

$$E_{r,c} = \dfrac{(\text{Sum of row } r) \times (\text{Sum of column } c)}{\text{Sample size}}$$

Question 1

Which of the following best describes a contingency table in the context of a chi-square test of independence?

Accepted Answer

A table that shows the observed frequencies for two categorical variables, arranged in rows and columns.

Question 2

Given a contingency table with 3 rows and 4 columns, what is the correct formula for the degrees of freedom for the chi-square test of independence?

Accepted Answer

$$df = (r - 1)(c - 1)$$

Question 3

A researcher collects data on students' preferred study method (group, solo, online) and their year in college (freshman, sophomore, junior, senior). Which statistical test is most appropriate to determine if study method preference is independent of year in college?

Accepted Answer

A chi-square test of independence

Question 4

Which of the following is the correct formula for the expected frequency $$E_{r,c} in a $$contingency table cell, assuming independence?

Accepted Answer

$$E_{r,c} = \dfrac{(	ext{Sum of row } r) 	imes (	ext{Sum of column } c)}{	ext{Sample size}}$$

Question 5

Which of the following is NOT a required condition for performing a chi-square test of independence?

Accepted Answer

The variables must be measured on an interval or ratio scale.

Question 6

Suppose you have a $$2 	imes 3$$ contingency table. What is the degrees of freedom for the chi-square test of independence?

Accepted Answer

$$df = 3$$

Question 7

A health club manager wants to know if the number of days per week college students exercise is related to gender. The null hypothesis for the chi-square test of independence is:

Accepted Answer

The number of days of exercise per week is independent of gender.

Question 8

Which of the following best describes the rejection region for a chi-square test of independence at significance level $$\alpha$$?

Accepted Answer

Values of $$\$$chi^2$$ greater than the critical value

Question 9

Given the following observed and expected frequencies for a cell: $$O = 40$$, $$E = 33.64. $$What is the contribution of this cell to the chi-square statistic?

Accepted Answer

$$\dfrac{(40 - 33.64)^2$}{33.64}$$

Question 10

A chi-square test of independence is performed and the test statistic is $$\$$chi^2$$ = 32.63$$ with $$df = 4$$ and $$\alpha = 0.01. $$The critical value is $$13.277. $$What is the correct conclusion?

Accepted Answer

Reject $$H_0$$; there is enough evidence to conclude the variables are dependent.

Question 11

Which of the following is a marginal frequency in a contingency table?

Accepted Answer

The total frequency for a row or column.

Question 12

In a chi-square test of independence, what does a large value of the test statistic $$\$$chi^2$$ indicate?

Accepted Answer

There is strong evidence against the null hypothesis of independence.

Question 13

Which of the following scenarios would violate the assumptions of the chi-square test of independence?

Accepted Answer

Some expected frequencies are less than 5.

Question 14

A researcher wants to test if favorite ice cream flavor is independent of age group. She collects data and finds that all expected frequencies are at least 5. What should she do next?

Accepted Answer

Calculate the test statistic using $$\$$chi^2$$ = \sum \dfrac{(O - E)^2$}{E}$$

Question 15

If the observed frequencies in a contingency table are very different from the expected frequencies, what is likely to happen in a chi-square test of independence?

Accepted Answer

The test statistic $$\$$chi^2$$ will be large, and H_0$ will likely be rejected.

Question 16

Which of the following best describes the null hypothesis $$H_0 in a $$chi-square test of independence?

Accepted Answer

The two categorical variables are independent.

Question 17

A chi-square test of independence is performed at $$\alpha = 0.05$$ with $$df = 3. $$The calculated $$\$$chi^2$$ statistic is 1.90. The critical value is 7.815. What is the correct interpretation?

Accepted Answer

Fail to reject $$H_0$$; there is not enough evidence to conclude dependence.

Question 18

Which of the following is a joint frequency in a contingency table?

Accepted Answer

The frequency in a specific cell at the intersection of a row and a column.

Question 19

In a real-world scenario, a hospital administrator wants to know if the type of treatment (A, B, or C) is independent of patient recovery status (Recovered, Not Recovered). Which statistical approach should be used?

Accepted Answer

Chi-square test of independence using a $$3 	imes 2$$ contingency table.

Chi-Square Tests for Independence: Contingency Tables and Statistical Inference

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