Chi-Square Test Questions: Expert Solutions

A researcher completes a chi-square test for independence and obtains ${\displaystyle {\chi }^2=6.2}$ for a sample of n=40 participants.
a. If the frequency data formed a ${\displaystyle 2\times 2}$ matrix, what is the phi-coefficient for the test?
b. If the frequency data formed a ${\displaystyle 3\times 3}$ matrix, what is the Cramer’s phi-coefficient for the test?
c. Explain why a very small value for an expected frequency can distort the results of chi-square test.

A recent study sought to determine if exercising after studying for an exam may help retain information. The 150 student volunteers that participated in the study were randomly assigned to three groups. All students were given a picture memory task that they studied for 15 minutes. After studying, one of the groups was instructed to exercise for 30 minutes immediately after studying. A second group was instructed to wait 1 hour after studying and then exercise for 30 minutes. The third group was instructed to not exercise after studying. The next day all 150 student volunteers were given a test to determine how well they could remember what they studied. Here are the results:
The researchers want to test: ${\displaystyle H_0}$: There is no difference in the distribution of performance on the test for the three treatment groups among the population of all student volunteers like these.
${\displaystyle H_{\alpha }}$: There is a difference in the distribution of performance on the test for the three treatment groups among the population of all student volunteers like these.
Which test is appropriate in this situation?
(A) chi-square test for homogeneity
(B) chi-square test for association
(C) chi-square test for goodness of fit
(D) two-sample test for p_1-p_2p1​-p2​
(E) matched pairs tt test

$\begin{array}{cc}& \text{Treatment Group}\\ \text{Performance on the Test}& \begin{array}{|ccccc|}\hline & \text{ Exercise immediately }& \text{ Exercise 1 hour later }& \text{ Do not exercise }& \text{ Total }\\ \text{ Less than 50\% }& 6& 8& 25& 39\\ \text{ 50\% to 75\% }& 22& 28& 22& 72\\ \text{ Greater than 75\% }& 22& 14& 3& 39\\ \text{ Total }& 50& 50& 50& 150\\ \hline\end{array}\end{array}$

A random sample of 500 people were classified by their ages into 3 age-groups: 29 years and younger, 30 to 64 years, and 65 years and older. Each person from the sample was surveyed about which of 4 major brands of cell phone they used. Their responses were compiled and displayed in a 3-by-4 contingency table. A researcher will use the data to investigate whether there is an association between cell phone brand and age- group. Which of the following is the appropriate test for the investigation?
a. A one-sample t-test for a population mean
b. A two-sample t-test for a difference between means
c. A chi-square goodness-of-fit test
d. A chi-square test of homogeneity
e. A chi-square test of independence

A political pollster selects a random sample of 75 registered voters from Allegheny County, Pennsylvania. The city of Pittsburgh is in Allegheny County. The pollster is interested in determining if the distribution of political party preference among registered voters in this county differs between those who live in Pittsburgh and those who live in the suburbs. The responses are displayed in the table.
$\begin{array}{cc}& \text{Political Party}\\ \text{Location}& \begin{array}{|cccc|}\hline & \text{ Democratic }& \text{ Independent }& \text{ Republican }\\ \text{ City }& 20& 8& 11\\ \text{ Suburbs }& 8& 10& 18\\ \hline\end{array}\end{array}$
The pollster would like to know if these data provide convincing evidence of a relationship between residential status and political party in the population of all registered voters in Allegheny County. What is the appropriate inference procedure?
1. chi-square test for goodness of fit because the data come from one random sample
2. chi-square test for homogeneity because the data come from independent random samples
3. chi-square test for association/independence because the data come from one random sample
4. chi-square test for association/independence because the data come from independent random samples

A flight attendant selects a random sample of 201 travelers. For each, he records the class in which the person travels and whether they have carry-on luggage. He would like to know if there is convincing evidence that the class in which a person travels is associated with having carry-on luggage. Let ${\displaystyle \alpha =0.05}$.
What are the hypotheses for this test?
a.${\displaystyle H_0}$: Class is not independent of carry-on luggage. ${\displaystyle H_{\alpha }}$: Class is independent of carry-on luggage.
b. ${\displaystyle H_0}$: Class is independent of carry-on luggage. ${\displaystyle H_{\alpha }}$: Class is not independent of carry-on luggage.
c. ${\displaystyle H_0}$: There is an association between class and carry-on luggage. ${\displaystyle H_{\alpha }}$: There is no association between class and carry-on luggage.
d. ${\displaystyle H_0}$: The proportion of travelers who have carry-on luggage is not the same in the types of class. ${\displaystyle H_{\alpha }}$: The proportion of travelers who have carry-on luggage is the same in the types of class.

A clothing manufacturer wants to determine whether customers prefer any specific color over other colors in shirts. He selects a random sample of 100 shirts sold and notes the color. The table below shows the result. At ${\displaystyle \propto =0.05}$, is there a color preference?
$\begin{array}{|cc|}\hline \text{ Color }& \text{ Number Sold }\\ \text{ White }& 43\\ \text{ Blue }& 22\\ \text{ Black }& 16\\ \text{ Red }& 9\\ \text{ Yellow }& 10\\ \hline\end{array}$
What test statistics will be used for this problem?
1. Chi-square test of Independence
2. Spearman rho
3. Chi-square test of goodness fit
4. Pearson
5. ANOVA

$\overline{\begin{array}{cc}& \text{Afraid to walk at night?}\\ & \text{Yes}& \text{No}& \text{Total}\\ \text{Male}& 173& 598& 771\\ \text{Female}& 393& 540& 933\\ \text{Total}& 566& 1138& 1704\\ \text{Source:}2014GSS\end{array}}$
If the chi-square $(\chi 2)$ test statistic $=73.7$ what is the p-value you would report? Use Table C.Remember to calculate the df first.
Group of answer choices
$P>0.250$
$P=0.01$
$P<0.001$
$P<0.002$

$\begin{array}{|cc|}\hline & \text{Right-Tail Probability}\\ df& 0.250& 0.100& 0.050& 0.025& 0.010& 0.005& 0.001\\ 1& 1.32& 2.71& 3.84& 5.02& 6.63& 7.88& 10.83\\ 2& 2.77& 4.61& 5.99& 7.38& 9.21& 10.60& 13.82\\ 3& 4.11& 6.25& 7.81& 9.35& 11.34& 12.84& 16.27\\ 4& 5.39& 7.78& 9.49& 11.14& 13.28& 14.86& 18.47\\ 5& 6.63& 9.24& 11.07& 12.83& 15.09& 16.75& 20.52\\ 6& 7.84& 10.64& 12.59& 14.45& 16.81& 18.55& 22.46\\ 7& 9.04& 12.02& 14.07& 16.01& 18.48& 20.28& 24.32\\ 8& 10.22& 13.36& 15.51& 17.53& 20.09& 21.96& 26.12\\ 9& 11.39& 14.68& 16.92& 19.02& 21.67& 23.59& 27.88\\ 10& 12.55& 15.99& 18.31& 20.48& 23.21& 25.19& 29.59\\ 11& 13.70& 17.28& 19.68& 21.92& 24.72& 26.76& 31.26\\ 12& 14.85& 18.55& 21.03& 23.34& 26.22& 28.30& 32.91\\ 13& 15.98& 19.81& 22.36& 24.74& 27.69& 29.82& 34.53\\ 14& 17.12& 21.06& 23.68& 26.12& 29.14& 31.32& 36.12\\ 15& 18.25& 22.31& 25.00& 27.49& 30.58& 32.80& 37.70\\ 16& 19.37& 32.54& 26.30& 28.85& 32.00& 34.27& 39.25\\ 17& 20.49& 24.77& 27.59& 30.19& 33.41& 35.72& 40.79\\ 18& 21.60& 25.99& 28.87& 31.53& 34.81& 37.16& 42.31\\ 19& 22.72& 27.20& 30.14& 32.85& 36.19& 38.58& 43.82\\ 20& 23.83& 28.41& 31.41& 34.17& 37.57& 40.00& 45.32\\ \hline\end{array}$

For each of the following situations, state whether you’d use a chi-square goodness-of-fit test, a chi-square test of homogeneity, a chi-square test of independence, or some other statistical test:
a) Is the quality of a car affected by what day it was built? A car manufacturer examines a random sample of the warranty claims filed over the past two years to test whether defects are randomly distributed across days of the work week.
b) A medical researcher wants to know if blood cholesterol level is related to heart disease. She examines a database of 10,000 patients, testing whether the cholesterol level (in milligrams) is related to whether or not a person has heart disease.
c) A student wants to find out whether political leaning (liberal, moderate, or conservative) is related to choice of major. He surveys 500 randomly chosen students and performs a test.

Explain the Chi – Square test.

Find the type and the number of variables for which the Chi-square goodness-of-fir tests are used.

The chi-square test is used when we have interval data.
A. True
B. False

Determine whether the chi-square test is applied to the categorical or the numerical data.

Suppose I gave you a bag of M&M’s, but I didn’t let you see the original packaging so you can’t determine which plant made the M& M’s. Your job is to count the number of candies of each color in your bag and figure out which plant made your bag. What test should you do to determine this?

For the following situations, identify the test you would run to analyze the data:
A marketing firm producing costumes is interested in studying consumer behavior in the context of purchase decision of costumes in a specific market. This company is a major player in the costume market that is characterized by intense competition. The company would like to know in particular whether the income level of the consumers (measured as lower, middle, upper middle, and upper class) influences their choice of costume type. They are specifically focused on four types of costumes (funny costumes, scary costumes, clever costumes, and boring costumes).
a. Chi-Square Goodness of Fit
b. Frequencies
c. Descriptive Statistics
d. Chi-Square of Independence

Chi-square tests are best used for which type of dependent variable?
nominal, ordinal
ordinal interval
nominal, interval
nominal, ratio

Determine whether the tests of proportion are used for the categorical or the numerical data.

State the null hypothesis for a chi-square test for independence.

To explain:Why the test for homogeneity follows the same procedures as the test for independence.

Describe a goodness-of-fit test. What assumptions are made when using a goodness-of-fit test?

A chi square test tells us:
a. Whether two continuous variables are correlated with each other.
b. Whether two discrete variables are independent of each other
c. The amount of variation in one variable explained by the other
d. Which categories of one variable are associated with which categories of the other variable