Q: How do you calculate the expected frequency for a cell in a contingency table?

$f_e = \frac{\text{row total} \times \text{column total}}{n}$

Question 1

Which of the following best describes a contingency table?

Accepted Answer

A table used to display the relationship between two or more categorical variables by classifying sample observations according to their characteristics.

Question 2

What is the formula for the chi-square test statistic used in contingency tables?

Accepted Answer

$\chi^2_{STAT} = \sum \frac{(f_o - f_e)^2}{f_e}$

Question 3

A university surveys 100 administrators, 50 students, and 50 faculty about a new policy. Which test should be used to determine if there are differences in the proportions of those who favor the policy among the three groups?

Accepted Answer

Chi-square test for differences among more than two proportions

Question 4

In a 2 x 2 contingency table, what is the degrees of freedom for the chi-square test statistic?

Accepted Answer

$1$

Question 5

Suppose you have the following observed and expected frequencies for a cell in a contingency table: $f_o = 20$, $f_e = 25$. What is the contribution of this cell to the chi-square statistic?

Accepted Answer

$\frac{(20-25)^2}{25} = 1$

Question 6

Which of the following is an assumption for the validity of the chi-square test in contingency tables?

Accepted Answer

Each expected cell frequency should be at least 1, and preferably at least 5.

Question 7

If the calculated chi-square statistic is less than the critical value at a given significance level, what is the correct decision?

Accepted Answer

Do not reject the null hypothesis.

Question 8

What is the null hypothesis for the chi-square test of independence?

Accepted Answer

The two categorical variables are independent.

Question 9

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

Accepted Answer

$(4-1) 	imes (3-1) = 6$

Question 10

How do you calculate the expected frequency for a cell in a contingency table?

Accepted Answer

$f_e = \frac{	ext{row total} 	imes 	ext{column total}}{n}$

Question 11

A chi-square test for independence is performed and the p-value is 0.9943. If $\alpha = 0.05$, what is the correct conclusion?

Accepted Answer

There is not enough evidence to conclude the variables are dependent.

Question 12

Which of the following scenarios is most appropriate for the Marascuilo procedure?

Accepted Answer

After rejecting the null hypothesis in a chi-square test for more than two proportions, to determine which specific pairs of proportions differ.

Question 13

What is the formula for the critical range in the Marascuilo procedure for comparing two proportions $p_j$ and $p_{j'}$?

Accepted Answer

$\sqrt{\chi^2_\alpha \left( \frac{p_j(1-p_j)}{n_j} + \frac{p_{j'}(1-p_{j'})}{n_{j'}} \right)}$

Question 14

A company wants to know if the type of meal plan (20/week, 10/week, none) is related to class standing (Freshman, Sophomore, Junior, Senior). Which statistical test should be used?

Accepted Answer

Chi-square test of independence

Question 15

If the absolute difference between two sample proportions $|p_j - p_{j'}|$ is greater than the critical range in the Marascuilo procedure, what does this indicate?

Accepted Answer

The two proportions are significantly different.

Question 16

Which of the following is NOT a correct use of the chi-square test?

Accepted Answer

Testing whether the mean of a population is equal to a specific value.

Question 17

A researcher finds $\chi^2_{STAT} = 12.792$ and the critical value $\chi^2_{0.01} = 9.2103$ for a test with 2 degrees of freedom. What is the correct conclusion at $\alpha = 0.01$?

Accepted Answer

Reject the null hypothesis; there is evidence of a difference among the groups.

Question 18

Which of the following best describes the null hypothesis for a chi-square test for the difference between two proportions?

Accepted Answer

$H_0: \pi_1 = \pi_2$

Question 19

In a real-world scenario, a marketing manager wants to know if customer satisfaction (satisfied, neutral, dissatisfied) is related to the region (North, South, East, West). What is the appropriate test and how many degrees of freedom does it have?

Accepted Answer

Chi-square test of independence with $(3-1)	imes(4-1) = 6$ degrees of freedom.

Chapter 11: Chi-Square Tests – Study Notes for Business Statistics

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