Chi-Square Test Questions: Expert Solutions

A doctor used to believe that babies were equally likely to be born any day of the week, but with the rise in scheduled deliveries, the doctor now claims that babies are twice as likely to be born on a weekday (Monday through Friday) than on a weekend (Saturday and Sunday). To test this theory, she selects a random sample of 120 babies born this year and determines on which day of the week they were born. She would like to carry out a test of significance to test her claim. What is the name of the appropriate inference procedure?
a. chi-square test for goodness of fit
b. one-sample t-test for a difference in means
c. two-sample t-test for a difference in means
d. two-sample z-test for a difference in proportions

A researcher obtains a negative value for the chi-square statistic in a chi-square goodness of fit test. What can a researcher conclude based on this result?
choices:
The observed frequencies are consistently larger than the expected frequencies.
The expected frequencies are consistently larger than the observed frequencies.
There are large differences between the observed and expected frequencies.
The researcher made a mistake because a chi-square value cannot be negative.

A researcher is interested in assessing if there are differences in the average number of monthly religious event attendances across religious affiliations. Religious event attendance is measured as an interval-ratio variable: it is the number of events that each survey respondent reported attending in the previous month. Religious affiliation is measured as a nominal variable with the following categories: Protestant, Catholic, Jewish, Muslim, Hindu, Buddhist, other, and not religious. Which of the following hypothesis tests would be appropriate, given the types of variables presented?
a. Chi-square test
b. Two-sample difference-of-proportions z-test
c. ANOVA
d. One-sample difference-of-proportions z-test

A random sample of 175 students was selected from a large high school. They were each asked their current grade level and whether they had a driver's license. The responses are displayed in the table.
$\begin{array}{cc}& \text{Grade Level}\\ \text{Driver's License?}& \begin{array}{|ccccc|}\hline & \text{ 9th }& \text{ 10th }& \text{ 11th }& \text{ 12th}\\ \text{ Yes }& 1& 5& 21& 48\\ \text{ No }& 28& 42& 20& 10\\ \hline\end{array}\end{array}$
The principal would like to know if these data provide convincing evidence of a relationship between grade level and having a driver's license in the population of all students in her school. 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 professor use chi-square test to examine the relationship between smoking and colon cancer. He followed a total of 480 people and the result was listed below:
$\begin{array}{|ccc|}\hline \text{ Frequency }& \text{ Colon Cancer }& \text{ No Colon Cancer }\\ \text{ Smokers }& 120& 80\\ \text{ Non-smokers }& 25& 255\\ \hline\end{array}$
a. 60.42
b. 101.20
c. 139.58
d. 84.58

A political' scientist wants to find out if voters are more likely to elect a senate candidate who opposes strict enforcement of immigration laws. She describes two candidates, each the same except that one supports strict enforcement while the other does not. She then asks a sample of voters which candidate they would vote for. The correct statistical test to use would be:
a. chi-square
b. paired samples t test
c. factorial ANOVA
d. one-way ANOVA

A manufacturer of cigarettes wishes to test the claim that the variance of the nicotine content of the cigarettes the company manufactures is equal to 0.638 milligram. The variance of a random sample of 25 cigarettes is 0.930 milligram. At ${\displaystyle \alpha =0.05}$, test the claim.

A large group of workers is divided into three departments: those who work in special operations, those who work on the main floor, and those who work in management. Strep throat has been going around and a health inspector would like to know if there is a relationship between department and having strep throat. She selects a random sample of 100 workers and classifies each one according to their department and whether they have strep throat. The data are displayed in the table.
The health inspector would like to know if these data provide convincing evidence that the distribution of strep throat status differs across the departments in the population of all workers of this large company. What is the appropriate inference procedure?
a. chi-square test for goodness of fit because the data come from one random sample
b. chi-square test for homogeneity because the data come from independent random samples
c. chi-square test for association/independence because the data come from one random sample
d. chi-square test for association/independence because the data come from independent random samples$\begin{array}{cc}& \text{Department}\\ \text{Strep?}& \begin{array}{|cccc|}\hline & \text{ Special Operations }& \text{ Main Floor }& \text{ Management }\\ \text{ Strep }& 6& 13& 5\\ \text{ No Strep }& 40& 30& 6\\ \hline\end{array}\end{array}$

A bullying prevention program is underway in the phoenix school districts. The researchers ran multiple analyses with the baseline data. The researchers want to determine if there is a relationship between age (13, 14, 15, 16, etc.) and victimization score (measured on a multi-item, Likert scale). What type of statistical test should be used?
Group of answer choices
Chi Square
Independent t-test
Correlation Analysis
Dependent t-test
Standard Deviation

A bicycle safety organization claims that fatal bicycle accidents are uniformly distributed throughout the week. The table on the right shows the day of the week for which 777 randomly selected fatal bicycle accidents occurred. At ${\displaystyle \alpha =0.01}$, can you Frequency, f Day Sunday Monday Tuesday Wednesday Thursday reject the claim that the distribution is uniform? Complete parts a through d below.
$\begin{array}{|cc|}\hline \text{ Day }& \text{ Frequency, f }\\ \text{ Sunday }& 107\\ \text{ Monday }& 113\\ \text{ Tuesday }& 105\\ \text{ Wednesday }& 110\\ \text{ Thursday }& 122\\ \text{ Friday }& 104\\ \text{ Saturday }& 116\\ \hline\end{array}$
Which hypothesis is the claim?
a. ${\displaystyle H_0}$
b. ${\displaystyle H_a}$
Determine the critical value, ${\displaystyle {\chi }_0^2}$ and the rejection region

A clinical trial is run to investigate the effectiveness of an experimental drug in reducing preterm delivery to a drug considered standard care and to placebo. Pregnant women are enrolled and randomly assigned to receive either the experimental drug, the standard drug or placebo. Women are followed through delivery and classified as delivering preterm (< 37 weeks) or not. The data are shown below. Is there a statistically significant difference in the proportions of women delivering preterm among the three treatment groups? Run the test at a 5% level of significance. Give each of the following
1) the appropriate null and alternative hypotheses; 2) the appropriate test; 3) the decision rule; 4) the calculation of the test statistic; and 5) your conclusion including a comparison to alpha or the critical value.
$\begin{array}{cc}& \text{Preterm Delivery}\\ & \begin{array}{|cccc|}\hline & \text{ Yes }& \text{ No }& \text{ Total }\\ \text{ Experimental Drug }& 7& 33& 40\\ \text{ Standard Drug }& 13& 32& 45\\ \text{ Placebo }& 17& 18& 35\\ \text{ Total }& 37& 83& 120\\ \hline\end{array}\end{array}$

A controversial issue in the sport of professional soccer is the use of instant replay for making difficult goal line decisions. Each person in a representative sample of 102 players, fans, coaches, and officials was asked his or her opinion about the use of instant replay for goal-line decisions. The data are summarized in the two-way frequency table below.
$\begin{array}{cc}& \text{Opinion}\\ \text{Category}& \begin{array}{|ccc|}\hline & \text{ Favor Use }& \text{ Oppose Use }\\ \text{ Players }& 22& 2\\ \text{ Fans }& 18& 6\\ \text{ Coaches }& 15& 26\\ \text{ Officials }& 3& 10\\ \hline\end{array}\end{array}$
In testing to see whether opinion with respect to the use of instant replay is independent of the category of the person interviewed, a chi-square test statistic of 27.99 and a p-value less than 0.001 were calculated. Which of the following statements is correct?
a. The number of degrees of freedom for the test is 8 - 1 = 7.
b. The chi-square test should not have been used because two of the counts in the table are less than 5.
c. The null hypothesis states that there is an association between category and opinion about the use of instant replay, and the small p-value suggests that the null hypothesis should be rejected.
d. The small p-value suggests that there is evidence of an association between category and opinion about the use of instant replay.
e. The chi-square test shows that fans favor the use of instant replay.

"Determine the critical values for these tests of a population standard deviation.
(a) A right-tailed test with 18 degrees of freedom at the ${\displaystyle \alpha =0.01}$ level of significance
(b) A left-tailed test for a sample of size n= 24 at the ${\displaystyle \alpha =0.1}$ level of significance
(c) A two-tailed test for a sample of size n= 26 at the ${\displaystyle \alpha =0.01}$ level of significance"
(a) The critical value for this right-tailed test is $◻$ (Round to three decimal places as needed.)
(b) The critical value for this left-tailed test is $◻$ ( Round to three decimal places as needed.)
(c) The critical values for this two-tailed test are $◻$ (Round to three decimal places as needed. Use a comma to separate answers as needed.)

A job candidate at a large job fair can be classified as unacceptable, provisional, or acceptable. Based on past experience, a high-quality candidate is expected to get 80 percent acceptable ratings, 15 percent provisional ratings, and 5 percent unacceptable ratings. A high-quality candidate was evaluated by 100 companies and received 60 acceptable, 25 provisional, and 15 unacceptable ratings. A chi-square goodness-of-fit-test was conducted to investigate whether the evaluation of the candidate is consistent with past experience. What is the value of the chi-square test statistic and number of degrees of freedom for the test?
a. ${\displaystyle {\chi }^2=\frac{{(15-5)}^2}{5}+\frac{{(25-15)}^2}{15}+\frac{{(60-80)}^2}{80}}$ with 2df
b. ${\displaystyle {\chi }^2=\frac{{(15-5)}^2}{5}+\frac{{(25-15)}^2}{15}+\frac{{(60-80)}^2}{80}}$ with 3df
c. ${\displaystyle {\chi }^2=\frac{{(15-5)}^2}{5}+\frac{{(25-15)}^2}{15}+\frac{{(60-80)}^2}{80}}$ with 99df
d. ${\displaystyle {\chi }^2=\frac{{(5-15)}^2}{15}+\frac{{(15-25)}^2}{25}+\frac{{(80-60)}^2}{60}}$ with 2df
e. ${\displaystyle {\chi }^2=\frac{{(5-15)}^2}{15}+\frac{{(15-25)}^2}{25}+\frac{{(80-60)}^2}{60}}$ with 3df

A manufacturing company with 350 employees is changing the employee health insurance plan to either plan A or plan B. The company wants to know if employees have a preference between the two plans and whether or not preference differs between those employees who have family members covered under the current plan (group 1) and those who do not (group 2). The human resources office takes a simple random sample from each of the two groups, sends information about both plans to the employees in each sample, and asks them whether they prefer plan A or plan B. The table summarizes the responses received, with expected cell counts in parentheses.
$\begin{array}{|cccc|}\hline & \text{ Plan A }& \text{ Plan B }& \text{ Total }\\ \text{ Yes (group 1) }& 40(32.5)& 20(27.5)& 60\\ \text{ No (group 2) }& 6(13.5)& 19(11.5)& 25\\ \text{ Total }& 46& 39& 85\\ \hline\end{array}$
Which statement is true about whether the conditions for the chi-square test for homogeneity have been met?
a. A simple random sample should have been taken from all the employees, and then each employee in the sample should have been asked their plan preference and whether or not they have family members covered under the current health insurance plan.
b. The expected cell counts are not large enough to apply the chi-square test for homogeneity.
c. The total sample size is not large enough to apply the chi-square test for homogeneity.
d. The total sample size is too large to apply the chi-square test for homogeneity.
e. All conditions necessary to apply the chi-square test for homogeneity are satisfied here.

1. Which of the following accurately describes the observed and expected frequencies in a chi-square test?
a. Both the observed and expected frequencies will always be whole numbers.
b. The observed frequencies are always whole numbers but the expected frequencies may be decimals or fractions.
c. The expected frequencies are always whole numbers but the observed frequencies may be decimals or fractions.
d. Both the observed and expected frequencies may be decimals or fractions.
2. What does it mean to obtain a negative value for the chi-square statistic?
a. The observed frequencies tend to be larger than the expected frequencies.
b. The expected frequencies tend to be larger than the observed frequencies.
c. There are large discrepancies between the observed and expected frequencies for most categories.
d. The chi-square statistic can never be negative.

A casino has been using six-sided dice obtained from Luckytown Dice Company for some of its games of chance. After a string of unusual rolls, the casino is concerned the dice are not fair (i.e. that distribution of rolls is not equally likely). They take a sample of 300 rolls. The observed counts are below. Calculate the value of the Chi-Square test statistic (Part of the calculations have been done for you). Please enter your answer to 2 decimal places.
$\begin{array}{|cccc|}\hline \text{ Value }& \text{ Observed }& \text{ Expected }& \text{ Contribution to Chi-square Test Statistic=(O-E)^2/E }\\ 1& 36& 50& (36-50{)}^2/50=3.92\\ 2& 48& 50& (48-50{)}^2/50=0.08\\ 3& 50& 50& (50-50{)}^2/50=0\\ 4& 43& 50& (43-50{)}^2/50=0.98\\ 5& 34& 50\\ 6& 89& 50\\ \text{ Total }& 300& 300\\ \hline\end{array}$

1) In using chi-square for testing the independence of variables, larger calculated values for chi-square will tend to make it more likely that the null hypothesis will be rejected.
True or False?
a) A contingency table for two nominal variables has three (3) row and four (4) columns. The number of degrees of freedom associated with the chi-square test of variable independence will be df= 6.
True or False?
b) In carrying out chi-square analysis, if one or more of the observed frequencies is less than five (5), rows or columns should be combined so that each value of observed frequency is at least 5.
True or False?

1. When the null hypothesis for a chi square test for independence is true.
a. a large difference between the observed frequencies and the expected frequencies.
b.little or no difference between the observed frequencies and the expected frequencies.
c. no difference between the observed frequencies and the marginals
d. no differences between the row and the column marginal
2. In the Chi Square test, degrees of freedom are calculated as :
a. N-1
b. ${\displaystyle N_1+N_2-2}$
c. (r+1)(c+1)
d. (r-1)(c-1)

1. For each of the following test z- test versus the t-test versus the chi squared test. Describe what the test is for and what conditions must be satisfied before test is conducted
2. Give at least one example of two variables that have positive correlation. Discuss why we would not be able to deduce causation in your example