Q: A researcher surveys 150 people about their preferred news source: TV, Internet, or Newspaper. The expected proportions are 0.5, 0.3, and 0.2. The observed counts are 70 (TV), 50 (Internet), and 30 (Newspaper). What is the expected count for Internet?

$$e_{Internet} = 150 \times 0.3 = 45$$

Q: A market researcher wants to test if the distribution of smartphone brands (Apple, Samsung, Other) among college students matches the national market share: 0.45, 0.35, 0.20. In a sample of 400 students, the observed counts are 180, 140, and 80. What is the expected count for 'Other'?

$$e_{Other} = 400 \times 0.20 = 80$$

Question 1

Which of the following scenarios is most appropriate for a chi-square goodness-of-fit test?

Accepted Answer

Testing whether the distribution of customer payment methods matches last year's proportions.

Question 2

A company wants to know if the distribution of preferred soda flavors (Cola, Lemon, Orange) among its customers has changed from last year, when the proportions were 0.4, 0.3, and 0.3, respectively. Which are the correct null and alternative hypotheses for a chi-square goodness-of-fit test?

Accepted Answer

$$H_0$$: The proportions are 0.4, 0.3, and 0.3; $$H_A$$: The proportions are not all 0.4, 0.3, and 0.3.

Question 3

Which formula correctly calculates the expected count $$e_i$$ for category $$i in a $$chi-square test?

Accepted Answer

$$e_i = n p_i$$

Question 4

A chi-square test statistic is calculated as $$\$$chi^2$$ = 12.5$$ with 3 degrees of freedom. If the critical value at $$\alpha = 0.05 is 7.815$$, what is the correct conclusion?

Accepted Answer

Reject $$H_0$$; there is evidence against the null hypothesis.

Question 5

In a contingency analysis with a 2x4 table (2 rows, 4 columns), what is the correct formula and value for the degrees of freedom?

Accepted Answer

$$df = (2-1)(4-1) = 3$$

Question 6

Which of the following is NOT an assumption required for a valid chi-square goodness-of-fit test?

Accepted Answer

The data are measured on an interval scale.

Question 7

A researcher surveys 150 people about their preferred news source: TV, Internet, or Newspaper. The expected proportions are 0.5, 0.3, and 0.2. The observed counts are 70 (TV), 50 (Internet), and 30 (Newspaper). What is the expected count for Internet?

Accepted Answer

$$e_{Internet} = 150 	imes 0.3 = 45$$

Question 8

Which Excel function would you use to find the critical value for a chi-square test with a right-tail probability of 0.05 and 4 degrees of freedom?

Accepted Answer

CHISQ.INV.RT(0.05, 4)

Question 9

A business analyst wants to test if the method of payment (Cash, Credit, Online) is independent of customer age group (Under 30, 30-50, Over 50). Which type of chi-square test should be used?

Accepted Answer

Chi-square test of independence (contingency analysis)

Question 10

If a chi-square test for independence yields a P-value of 0.12 and the significance level is $$\alpha = 0.05$$, what is the correct interpretation?

Accepted Answer

Fail to reject $$H_0$$; there is not enough evidence to conclude the variables are associated.

Question 11

A chi-square test is performed with 5 categories and 2 parameters estimated from the data. What is the correct formula and value for the degrees of freedom?

Accepted Answer

$$df = 5 - 2 - 1 = 2$$

Question 12

Which of the following best describes the chi-square test statistic?

Accepted Answer

$$\$$chi^2$$ = \sum \frac{(o_i - e_i)^2$}{e_i}$$

Question 13

Which of the following is a correct interpretation of a large chi-square statistic?

Accepted Answer

There is a large difference between observed and expected counts, suggesting incompatibility with the null hypothesis.

Question 14

A market researcher wants to test if the distribution of smartphone brands (Apple, Samsung, Other) among college students matches the national market share: 0.45, 0.35, 0.20. In a sample of 400 students, the observed counts are 180, 140, and 80. What is the expected count for 'Other'?

Accepted Answer

$$e_{Other} = 400 	imes 0.20 = 80$$

Question 15

Which of the following is a valid reason to use a chi-square test of independence?

Accepted Answer

To determine if two categorical variables, such as gender and product preference, are related.

Question 16

A chi-square test is performed with 4 categories and no parameters estimated from the data. What is the degrees of freedom?

Accepted Answer

$$df = 4 - 1 = 3$$

Question 17

Which of the following is a correct data condition for using the chi-square test in contingency analysis?

Accepted Answer

All expected counts in the contingency table are at least 5.

Question 18

A company wants to test if the distribution of customer complaints (Product, Service, Delivery) has changed after a new policy. Last year, the proportions were 0.5, 0.3, 0.2. This year, in a sample of 250 complaints, the observed counts are 120, 80, and 50. What is the chi-square test statistic?

Accepted Answer

Question 19

A chi-square test is performed with observed counts [54, 48, 98] and expected counts [40, 60, 100]. What is the value of the test statistic?

Accepted Answer

$$\$$chi^2$$ = \frac{(54-40)^2$}{40} + \frac{(48-60)^2$}{60} + \frac{(98-100)^2$}{100} = \frac{196}{40} + \frac{144}{60} + \frac{4}{100} = 4.90 + 2.40 + 0.04 = 7.34$$

Chi-square Tests: Goodness-of-Fit and Contingency Analysis

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