Q: A sociologist wants to test if the distribution of years grandparents care for their grandchildren has changed since 2000. The expected counts for 1,000 grandparents are calculated using $$E_i = n p_i. If$$ $$p_{3-4yr} = 0.176$$, what is the expected count for the '3 or 4 years' category?

$$E_{3-4yr} = 1000 \times 0.176 = 176$$

Q: Given the contingency table below for a survey of 883 men and 893 women about their preference for money, health, or love, what is the expected count for men who chose 'Health'? (Row total for men: 883, column total for 'Health': 1020, table total: 1776)

$$E = \frac{883 \times 1020}{1776} \approx 507.05$$

Question 1

Which of the following statements about the chi-square distribution is TRUE?

Accepted Answer

As the degrees of freedom increase, the distribution becomes more symmetric.

Question 2

A sociologist wants to test if the distribution of years grandparents care for their grandchildren has changed since 2000. The expected counts for 1,000 grandparents are calculated using $$E_i = n p_i. If$$ $$p_{3-4yr} = 0.176$$, what is the expected count for the '3 or 4 years' category?

Accepted Answer

$$E_{3-4yr} = 1000 	imes 0.176 = 176$$

Question 3

Which formula is used to calculate the chi-square test statistic for a goodness-of-fit test?

Accepted Answer

$$\$$chi^2$$ = \sum_{i=1}^k \frac{(O_i - E_i)^2$}{E_i}$$

Question 4

In a chi-square goodness-of-fit test, which of the following is a requirement for the expected counts?

Accepted Answer

No more than 20% of expected counts are less than 5.

Question 5

Suppose you conduct a chi-square goodness-of-fit test with $$k = 4$$ categories and a significance level $$\alpha = 0.05. If $$your test statistic is $$\$$chi^2$$_0 = 6.605$$ and the critical value is $$\$$chi^2$$_{0.05} = 7.815$$, what is the correct conclusion?

Accepted Answer

Fail to reject the null hypothesis because $$6.605 \u003c 7.815.$$

Question 6

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

Accepted Answer

The two variables are independent.

Question 7

Given the contingency table below for a survey of 883 men and 893 women about their preference for money, health, or love, what is the expected count for men who chose 'Health'? (Row total for men: 883, column total for 'Health': 1020, table total: 1776)

Accepted Answer

$$E = \frac{883 	imes 1020}{1776} \approx 507.05$$

Question 8

Which of the following is the correct formula for the degrees of freedom in a chi-square test for independence with $$r$$ rows and $$c$$ columns?

Accepted Answer

$$(r-1)(c-1)$$

Question 9

A chi-square test for homogeneity of proportions is used to:

Accepted Answer

Test whether different populations have the same proportion of individuals with a characteristic.

Question 10

In a chi-square test, if the P-value is less than the significance level $$\alpha$$, what is the correct decision?

Accepted Answer

Reject the null hypothesis.

Question 11

A researcher wants to test if the proportion of people who believe teaching is a prestigious career is the same in 1992, 2002, and 2008. The null hypothesis is:

Accepted Answer

$$H_0$$: $$p_1$$ = $$p_2$$ = $$p_3$$

Question 12

Which of the following is NOT a valid reason to combine categories in a chi-square test?

Accepted Answer

To artificially increase the test statistic.

Question 13

If a chi-square test statistic is much larger than the critical value, what does this indicate about the observed and expected counts?

Accepted Answer

There is a significant difference between observed and expected counts.

Question 14

Which of the following best describes the area to the right of the test statistic in a chi-square distribution?

Accepted Answer

The P-value

Question 15

A chi-square test for independence is performed on a $$2 	imes 3$$ contingency table. What are the degrees of freedom?

Accepted Answer

$$(2-1)(3-1) = 2$$

Question 16

Which of the following is a correct interpretation if you fail to reject the null hypothesis in a chi-square goodness-of-fit test?

Accepted Answer

There is insufficient evidence to conclude the distribution is different from the hypothesized distribution.

Chi-Square Tests: Goodness-of-Fit, Independence, and Homogeneity

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