Deck 13: Chi-Square Tests

Question 1

A multinomial probability distribution describes data that are classified into two or more categories when a multinomial experiment is carried out.

Answer 1

True

Question 2

A fastener manufacturing company uses a chi-square goodness-of-fit test to determine if a population of all lengths of ¼-inch bolts it manufactures is distributed according to a normal distribution. If we reject the null hypothesis, it is reasonable to assume that the population distribution is approximately normally distributed.

Answer 2

False

Question 3

The chi-square goodness-of-fit test can only be used to test whether a population has specified multinomial probabilities or to test if a sample has been selected from a normally distributed population. It cannot be used if sample data come from other distribution forms, such as the Poisson.

Answer 3

False

Question 4

When we carry out a chi-square test of independence, the expected frequencies are based on the null hypothesis.

Question 5

The χ² goodness-of-fit test requires the nominative level of data.

Question 6

One use of the chi-square goodness-of-fit test is to determine if specified multinomial probabilities in the null hypothesis are favored over the alternative hypothesis.

Question 7

In a contingency table, if all of the expected frequencies equal the observed frequencies, then we can conclude that there is a perfect association between rows and columns.

Question 8

The actual counts in the cells of a contingency table are referred to as the expected cell frequencies.

Question 9

When we carry out a chi-square test of independence, in the alternative hypothesis we state that the two classifications are statistically independent.

Question 10

When using a chi-square goodness-of-fit test with multinomial probabilities, the rejection of the null hypothesis indicates that at least one of the multinomial probabilities is not equal to the value stated in the null hypothesis.

Question 11

In a contingency table, when all the expected frequencies equal the observed frequencies, the calculated χ² statistic equals zero.

Question 12

A contingency table summarizes data that has been classified into two dimensions or scales.

Question 13

The trials of a multinomial probability are assumed to be dependent.

Question 14

In performing a chi-square test of independence, as the difference between the respective observed and expected frequencies calculated by assuming independence decreases, the probability of concluding that the row variable is independent of the column variable decreases.

Question 15

In performing a chi-square goodness-of-fit test with multinomial probabilities, the smaller the difference between observed and expected frequencies, the higher the probability of concluding that the probabilities specified in the null hypothesis are favored over the alternative hypothesis.

Question 16

When using the chi-square goodness-of-fit test, if the value of the chi-square statistic is large enough, we reject the null hypothesis.

Question 17

The chi-square distribution is a continuous probability distribution that is skewed to the left.

Question 18

When we carry out a chi-square test of independence, if r_i is the row total for row i and c_j is the column total for column j, then the estimated expected cell frequency corresponding to row i and column j equals (r_i)(c_j)/n.

Question 19

When we carry out a chi-square test of independence, the chi-square statistic is based on degrees of freedom, where r and c denote, respectively, the number of rows and columns in the contingency table.

Question 20

Expected cell frequencies for a multinomial distribution are calculated by assuming statistical dependence.

Question 21

The chi-square goodness-of-fit is ________ a one-tailed test with the rejection region in the right tail.

A) always
B) sometimes
C) never

Question 22

When we carry out a chi-square goodness-of-fit test for a normal distribution, the null hypothesis states that the population

A) does not have a normal distribution.
B) has a normal distribution.
C) has a chi-square distribution.
D) does not have a chi-square distribution.
E) has k − 3 degrees of freedom.

Question 23

As the difference between observed frequency and expected frequency ________, the probability of rejecting the null hypothesis increases.

A) stays the same
B) decreases
C) increases
D) goes to 0

Question 24

The number of degrees of freedom associated with a chi-square test for independence based upon a contingency table with 4 rows and 3 columns is ________.

A) 7
B) 12
C) 5
D) 6

Question 25

When we carry out a chi-square test of independence, the alternate hypothesis states that the two relevant classifications

A) are mutually exclusive.
B) form a contingency table with r rows and c columns.
C) have (r − 1)(c − 1) degrees of freedom.
D) are statistically dependent.
E) are normally distributed.

Question 26

In performing a chi-square goodness-of-fit test for a normal distribution, a researcher wants to make sure that all of the expected cell frequencies are at least five. The sample is divided into 7 intervals. The second through the sixth intervals all have expected cell frequencies of at least five. The first and the last intervals have expected cell frequencies of 1.5 each. After adjusting the number of intervals, the degrees of freedom for the chi-square statistic is ________.

A) 2
B) 3
C) 5
D) 7

Question 27

While a binomial distribution describes count data that can be classified into one of two mutually exclusive categories, a ________ distribution describes count data that are classified into more than two mutually exclusive categories.

A) normal
B) skewed
C) uniform
D) multinomial

Question 28

The χ² statistic is used to test whether the assumption of normality is reasonable for a given population distribution. The sample consists of 5,000 observations and is divided into 6 categories (intervals). The degrees of freedom for the chi-square statistic are

A) 4,999.
B) 6.
C) 5.
D) 4.
E) 3.

Question 29

A special version of the chi-square goodness-of-fit test that involves testing the null hypothesis that all of the multinomial probabilities are equal is called the test for ________.

A) goodness of fit
B) statistical independence
C) normality
D) homogeneity

Question 30

In performing a chi-square goodness-of-fit test for a normal distribution, if there are 7 intervals, then the number of degrees of freedom for the chi-square statistic is ________.

A) 7
B) 3
C) 4
D) 6

Question 31

The χ² statistic from a contingency table with 6 rows and 5 columns will have

A) 30 degrees of freedom.
B) 24 degrees of freedom.
C) 5 degrees of freedom.
D) 20 degrees of freedom.
E) 25 degrees of freedom.

Question 32

In performing a chi-square test of independence, as the differences between respective observed and expected frequencies ________, the probability of concluding that the row variable is independent of the column variable increases.

A) stay the same
B) decrease
C) increase
D) double

Question 33

A manufacturing company produces part QV2Y for the aerospace industry. This particular part can be manufactured using 3 different production processes. The management wants to know if the quality of the units of part QV2Y is the same for all three processes. The production supervisor obtained the following data: Process 1 had 29 defective units in 240 items, Process 2 produced 12 defective units in 180 items, and Process 3 manufactured 9 defective units in 150 items. At a significance level of 0.05, we performed a chi-square test to determine whether the quality of the items produced appears to be the same for all three processes. What is the null hypothesis?

A) H₀: The number of defectives produced is independent of the production process used.
B) H₀: The row and column variables are associated with each other.
C) H₀: The proportion of defective units produced by the three production processes is the same.
D) Both "H₀: The number of defectives produced is independent of the production process used." and "H₀: The proportion of defective units produced by the three production processes is the same." are correct or at least acceptable ways of stating the null hypothesis.
E) All of the other choices are acceptable ways of stating the null hypothesis.

Question 34

Which, if any, of the following statements about the chi-square test of independence is false?

A) If r_i is the row total for row i and c_j is the column total for column j, then the estimated expected cell frequency corresponding to row i and column j equals (r_i)(c_j)/n.
B) The test is valid if all of the estimated cell frequencies are at least five.
C) The chi-square statistic is based on (r − 1)(c − 1) degrees of freedom, where r and c denote, respectively, the number of rows and columns in the contingency table.
D) The alternative hypothesis states that the two classifications are statistically independent.
E) All of the other statements about the chi-square test of independence are true.

Question 35

The chi-square goodness-of-fit test will be valid if the average of all of the expected cell frequencies is ________.

A) greater than 0
B) less than 5
C) between 0 and 5
D) at least 1
E) at least 5

Question 36

When we carry out a chi-square test of independence, as the differences between the respective observed and expected frequencies decrease, the probability of concluding that the row variable is independent of the column variable

A) decreases.
B) increases.
C) may decrease or increase depending on the number of rows and columns.
D) will be unaffected.

Question 37

An experiment consists of 400 observations and four mutually exclusive groups. If the probability of a randomly selected item being classified into any of the four groups is equal, then the expected number of items that will be classified into group 1 is ________.

A) 25
B) 100
C) 125
D) 150

Question 38

In performing a chi-square goodness-of-fit test with multinomial probabilities, the ________ the difference between observed and expected frequencies, the higher the probability of concluding that the probabilities specified in the null hypothesis are favored over the alternative hypothesis.

A) larger
B) smaller

Question 39

The chi-square goodness-of-fit test for multinomial probabilities with 5 categories has ________ degrees of freedom.

A) 5
B) 4
C) 3
D) 6

Question 40

When we carry out a goodness-of-fit chi-square test, the expected frequencies are based on the alternative hypothesis.

Question 41

A manufacturing company produces part QV2Y for the aerospace industry. This particular part can be manufactured using 3 different production processes. The management wants to know if the quality of the units of part QV2Y is the same for all three processes. The production supervisor obtained the following data: Process 1 had 29 defective units in 240 items, Process 2 produced 12 defective units in 180 items, and Process 3 manufactured 9 defective units in 150 items. At a significance level of .05, the management wants to perform a hypothesis test to determine whether the quality of items produced appears to be independent of the production process used. What is the rejection point condition?

A) Reject H₀ if χ² > .10257
B) Reject H₀ if χ² > 9.3484
C) Reject H₀ if χ² > 5.99147
D) Reject H₀ if χ² > 7.37776
E) Reject H₀ if χ²> 7.81473

Question 42

Consider the 3 × 2 contingency table below.
Compute the expected frequencies in row 3.

Question 43

A manufacturing company produces part QV2Y for the aerospace industry. This particular part can be manufactured using 3 different production processes. The management wants to know if the quality of the units of part QV2Y is the same for all three processes. The production supervisor obtained the following data: Process 1 had 29 defective units in 240 items, Process 2 produced 12 defective units in 180 items, and Process 3 manufactured 9 defective units in 150 items. Chi-Square Contingency Table Test for Independence
At a significance level of 0.05, the management wants to perform a hypothesis test to determine if the quality of the items produced appears to be independent of the production process used. Based on the results summarized in the MegaStat/Excel output provided in the table above, we

A) reject H₀ and conclude that the quality of the product is not the same for all processes.
B) reject H₀ and conclude that the quality of the product is dependent on the manufacturing process.
C) do not reject H₀, and conclude that the quality of the product does not significantly differ among the three processes.
D) do not reject H₀, and conclude that the quality of the product is not the same for all processes.
E) reject H₀ and conclude that the quality of the product is independent of the manufacturing process used.

Question 44

Consider a set of 50 measurements with mean 50.2 and standard deviation 18.7 and with the following observed frequencies.
It is desired to test whether these measurements came from a normal population. Calculate the expected frequency for the interval 0 − 39.99.

Question 45

Consider the 3 × 2 contingency table below.
How many degrees of freedom are associated with the chi-square test?

Question 46

A manufacturing company produces part QV2Y for the aerospace industry. This particular part can be manufactured using 3 different production processes. The management wants to know if the quality of the units of part QV2Y is the same for all three processes. The production supervisor obtained the following data: Process 1 had 29 defective units in 240 items, Process 2 produced 12 defective units in 180 items, and Process 3 manufactured 9 defective units in 150 items. At a significance level of 0.05, we performed a chi-square test of independence to determine if the quality of the items produced appears to be independent of the production process. What are the degrees of freedom for the chi-square statistic?

A) 2
B) 3
C) 50
D) 520
E) 570

Question 47

A real estate company is analyzing the selling prices of residential homes in a given community. 140 homes that have been sold in the past month are randomly selected and their selling prices are recorded. The statistician working on the project has stated that in order to perform various statistical tests, the data must be distributed according to a normal distribution. In order to determine whether the selling prices of homes included in the random sample are normally distributed, the statistician divides the data into 6 classes of equal size and records the number of observations in each class. She then performs a chi-square goodness-of-fit test for normal distribution. The results are summarized in the following table. Goodness-of-Fit Test
At a significance level of .05, what is the appropriate rejection point condition?

A) Reject H₀ if χ² > 12.5916
B) Reject H₀ if χ² > 11.0705
C) Reject H₀ if χ² > 9.3484
D) Reject H₀ if χ² > 7.81473
E) Reject H₀ if χ² > 9.48773

Question 48

A real estate company is analyzing the selling prices of residential homes in a given community. 140 homes that have been sold in the past month are randomly selected and their selling prices are recorded. The statistician working on the project has stated that in order to perform various statistical tests, the data must be distributed according to a normal distribution. In order to determine whether the selling prices of homes included in the random sample are normally distributed, the statistician divides the data into 6 classes of equal size and records the number of observations in each class. She then performs a chi-square goodness-of-fit test for normal distribution. The results are summarized in the following table. Goodness-of-Fit Test
What is the appropriate null hypothesis?

A) H₀: The residential home selling prices are distributed according to a normal distribution.
B) H₀: The residential home selling prices are not distributed according to a normal distribution.
C) H₀: The distribution of residential home selling prices is either right or left skewed.
D) H₀: The distribution of the residential home selling prices is symmetric.
E) None of the other answers is correct.

Question 49

A real estate company is analyzing the selling prices of residential homes in a given community. 140 homes that have been sold in the past month are randomly selected and their selling prices are recorded. The statistician working on the project has stated that in order to perform various statistical tests, the data must be distributed according to a normal distribution. In order to determine whether the selling prices of homes included in the random sample are normally distributed, the statistician divides the data into 6 classes of equal size and records the number of observations in each class. She then performs a chi-square goodness-of-fit test for normal distribution. The results are summarized in the following table. Goodness-of-Fit Test
At a significance level of .05, we

A) reject H₀; conclude that the residential home selling prices are not distributed according to a normal distribution.
B) do not reject H₀; conclude that the residential home selling prices are not distributed according to a normal distribution.
C) reject H₀; conclude that the residential home selling prices are distributed according to a normal distribution.
D) do not reject H₀; conclude that the residential home selling prices are distributed according to a normal distribution.

Question 50

Consider a set of 50 measurements with mean 50.2 and standard deviation 18.7 and with the following observed frequencies.
It is desired to test whether these measurements came from a normal population. Calculate the expected frequency for the interval 80 and higher.

Question 51

Consider a set of 50 measurements with mean 50.2 and standard deviation 18.7 and with the following observed frequencies.
It is desired to test whether these measurements came from a normal population. Calculate the expected frequency for the interval 60 - 79.99.

Question 52

Consider the 3 × 2 contingency table below.
At α = .05, determine the tabular value of the chi-square statistic used to test for the independence of factors A and B.

Question 53

Consider the 3 × 2 contingency table below.
Compute the expected frequencies in row 1.

Question 54

Consider the 3 × 2 contingency table below.
At a significance level of .05, test H₀: the factors A and B are independent.

Question 55

A real estate company is analyzing the selling prices of residential homes in a given community. 140 homes that have been sold in the past month are randomly selected and their selling prices are recorded. The statistician working on the project has stated that in order to perform various statistical tests, the data must be distributed according to a normal distribution. In order to determine whether the selling prices of homes included in the random sample are normally distributed, the statistician divides the data into 6 classes of equal size and records the number of observations in each class. She then performs a chi-square goodness-of-fit test for normal distribution. The results are summarized in the following table Goodness-of-Fit Test
What are the degrees of freedom for the chi-square test?

A) 2
B) 3
C) 4
D) 5
E) 6

Question 56

A manufacturing company produces part QV2Y for the aerospace industry. This particular part can be manufactured using 3 different production processes. The management wants to know if the quality of the units of part QV2Y is the same for all three processes. The production supervisor obtained the following data: Process 1 had 29 defective units in 240 items, Process 2 produced 12 defective units in 180 items, and Process 3 manufactured 9 defective units in 150 items. Chi-Square Contingency Table Test for Independence
At a significance level of .10, the management wants to perform a hypothesis test to determine if the quality of the items produced appears to be independent of the production process used. Based on the results summarized in the MegaStat/Excel output provided in the table above, we

A) do not reject H₀ and conclude that the quality of the product is not the same for all processes.
B) reject H₀ and conclude that the quality of the product is dependent on the manufacturing process.
C) do not reject H₀, and conclude that the quality of the product does not significantly differ among the three processes.
D) reject H₀ and conclude that the quality of the product is independent of the production process utilized.

Question 57

Consider the 3 × 2 contingency table below.
Compute the expected frequencies in row 2.

Question 58

Consider a set of 50 measurements with mean 50.2 and standard deviation 18.7 and with the following observed frequencies.
It is desired to test whether these measurements came from a normal population. How many degrees of freedom are associated with the chi-square test?

Question 59

Consider a set of 50 measurements with mean 50.2 and standard deviation 18.7 and with the following observed and expected frequencies.
It is desired to test whether these measurements came from a normal population. Calculate the value of the chi-square test statistic.

Question 60

Consider a set of 50 measurements with mean 50.2 and standard deviation 18.7 and with the following observed frequencies.
It is desired to test whether these measurements came from a normal population. Calculate the expected frequency for the interval 40 − 59.99.

Question 61

In the past, of all the students enrolled in Basic Business Statistics, 10 percent earned an A, 20 percent earned a B, 30 percent earned a C, 20 percent earned a D, and the remainder failed the course. Dr. Johnson is a new professor teaching Basic Business Statistics for the first time this semester. At the conclusion of the semester, of his 60 students, 10 had earned an A, 20 a B, 20 a C, 5 a D, and 5 received an F. Assume that the class constitutes a random sample. Dr. Johnson wants to know if there is sufficient evidence to conclude that the grade distribution of his class is different from the historical grade distribution. At α = .05, test to determine if the grade distribution for this class is different from the historical grade distribution.

Question 62

On the most recent tax cut proposal, a random sample of Democrats and Republicans in the Congress cast their votes as follows.
Suppose that the chi-square test of independence is performed and the null hypothesis (the vote on the issue and party affiliation are independent) is rejected. Provide a one-sentence interpretation of the outcome of the test.

Question 63

A U.S.-based company offers an online proficiency course in basic accounting. Completing this online course satisfies the Fundamentals of Accounting course requirement in many MBA programs. In the first semester, 315 students have enrolled in the course. The marketing research manager divided the country into seven regions of approximately equal population. The course enrollment values for each of the seven regions are given below. The management wants to know if there is equal interest in the course across all regions.
At a significance level of .05, test H₀: the probabilities are equal for all seven regions.

Question 64

In the past, of all the students enrolled in Basic Business Statistics, 10 percent earned an A, 20 percent earned a B, 30 percent earned a C, 20 percent earned a D, and the remainder failed the course. Dr. Johnson is a new professor teaching Basic Business Statistics for the first time this semester. At the conclusion of the semester, of his 60 students, 10 had earned an A, 20 a B, 20 a C, 5 a D, and 5 received an F. Assume that the class constitutes a random sample. Dr. Johnson wants to know if there is sufficient evidence to conclude that the grade distribution of his class is different from the historical grade distribution. Use α = .05 and determine the appropriate degrees of freedom and the rejection point condition associated with this goodness-of-fit test.

Question 65

In the past, of all the students enrolled in Basic Business Statistics, 10 percent earned an A, 20 percent earned a B, 30 percent earned a C, 20 percent earned a D, and the remainder failed the course. Dr. Johnson is a new professor teaching Basic Business Statistics for the first time this semester. At the conclusion of the semester, of his 60 students, 10 had earned an A, 20 a B, 20 a C, 5 a D, and 5 received an F. Assume that the class constitutes a random sample. Dr. Johnson wants to know if there is sufficient evidence to conclude that the grade distribution of his class is different from the historical grade distribution. If we assume at α = .05 and that the null hypothesis is rejected, make a one-sentence managerial conclusion.

Question 66

A U.S.-based company offers an online proficiency course in basic accounting. Completing this online course satisfies the Fundamentals of Accounting course requirement in many MBA programs. In the first semester, 315 students have enrolled in the course. The marketing research manager divided the country into seven regions of approximately equal population. The course enrollment values for each of the seven regions are given below. The management wants to know if there is equal interest in the course across all regions.
How many degrees of freedom are associated with the chi-square test? Also, at α = 0.05, determine the rejection point condition of the chi-square statistic.

Question 67

In the past, of all the students enrolled in Basic Business Statistics, 10 percent earned an A, 20 percent earned a B, 30 percent earned a C, 20 percent earned a D, and the remainder failed the course. Dr. Johnson is a new professor teaching Basic Business Statistics for the first time this semester. At the conclusion of the semester, of his 60 students, 10 had earned an A, 20 a B, 20 a C, 5 a D, and 5 received an F. Assume that the class constitutes a random sample. Dr. Johnson wants to know if there is sufficient evidence to conclude that the grade distribution of his class is different from the historical grade distribution. Calculate the expected values for an A and for a D.

Question 68

On the most recent tax cut proposal, a random sample of Democrats and Republicans in the Congress cast their votes as follows.
Determine the expected frequencies for both the Democrats and Republicans who oppose the tax cut proposal for the chi-square test of independence.

Question 69

A U.S.-based company offers an online proficiency course in basic accounting. Completing this online course satisfies the Fundamentals of Accounting course requirement in many MBA programs. In the first semester, 315 students have enrolled in the course. The marketing research manager divided the country into seven regions of approximately equal population. The course enrollment values for each of the seven regions are given below. The management wants to know if there is equal interest in the course across all regions.
Assume that H₀: p₁ = p₂ = p₃ = p₄ = p₅ = p₆ = p₇ is not rejected, and state a one-sentence managerial conclusion.

Question 70

In the past, of all the students enrolled in Basic Business Statistics, 10 percent earned an A, 20 percent earned a B, 30 percent earned a C, 20 percent earned a D, and the remainder failed the course. Dr. Johnson is a new professor teaching Basic Business Statistics for the first time this semester. At the conclusion of the semester, of his 60 students, 10 had earned an A, 20 a B, 20 a C, 5 a D, and 5 received an F. Assume that the class constitutes a random sample. Dr. Johnson wants to know if there is sufficient evidence to conclude that the grade distribution of his class is different from the historical grade distribution. Calculate the chi-square statistic.

Question 71

On the most recent tax cut proposal, a random sample of Democrats and Republicans in the Congress cast their votes as follows.
At a significance level of .01, determine the appropriate degrees of freedom and the rejection point condition for this test.

Question 72

A U.S.-based company offers an online proficiency course in basic accounting. Completing this online course satisfies the Fundamentals of Accounting course requirement in many MBA programs. In the first semester, 315 students have enrolled in the course. The marketing research manager divided the country into seven regions of approximately equal population. The course enrollment values for each of the seven regions are given below. The management wants to know if there is equal interest in the course across all regions.
Assume that H₀, the probabilities are equal for all seven regions, is rejected. State a one-sentence managerial conclusion.

Question 73

In the past, of all the students enrolled in Basic Business Statistics, 10 percent earned an A, 20 percent earned a B, 30 percent earned a C, 20 percent earned a D, and the remainder failed the course. Dr. Johnson is a new professor teaching Basic Business Statistics for the first time this semester. At the conclusion of the semester, of his 60 students, 10 had earned an A, 20 a B, 20 a C, 5 a D, and 5 received an F. Assume that the class constitutes a random sample. Dr. Johnson wants to know if there is sufficient evidence to conclude that the grade distribution of his class is different from the historical grade distribution. Calculate the expected values for a B and for a C.

Question 74

A U.S.-based company offers an online proficiency course in basic accounting. Completing this online course satisfies the Fundamentals of Accounting course requirement in many MBA programs. In the first semester, 315 students have enrolled in the course. The marketing research manager divided the country into seven regions of approximately equal population. The course enrollment values for each of the seven regions are given below. The management wants to know if there is equal interest in the course across all regions.
Calculate the value of the chi-square statistic.

Question 75

A U.S.-based company offers an online proficiency course in basic accounting. Completing this online course satisfies the Fundamentals of Accounting course requirement in many MBA programs. In the first semester, 315 students have enrolled in the course. The marketing research manager divided the country into seven regions of approximately equal population. The course enrollment values for each of the seven regions are given below. The management wants to know if there is equal interest in the course across all regions.
Calculate the expected enrollment (frequency) for all 7 regions.

Question 76

Consider a set of 50 measurements with mean 50.2 and standard deviation 18.7 and with the following observed and expected frequencies.
It is desired to test whether these measurements came from a normal population. At a significance level of .05, test H₀: the set of 50 measurements came from a normal population.

Question 77

On the most recent tax cut proposal, a random sample of Democrats and Republicans in the Congress cast their votes as follows.
Use a significance level of .01 and determine whether the opinions on the tax cut proposal and the party affiliation are independent.

Question 78

On the most recent tax cut proposal, a random sample of Democrats and Republicans in the Congress cast their votes as follows.
Calculate the chi-square statistic for this test of independence.

Question 79

A U.S.-based company offers an online proficiency course in basic accounting. Completing this online course satisfies the Fundamentals of Accounting course requirement in many MBA programs. In the first semester, 315 students have enrolled in the course. The marketing research manager divided the country into seven regions of approximately equal population. The course enrollment values for each of the seven regions are given below. The management wants to know if there is equal interest in the course across all regions.
At a significance level of .01, test H₀: the probabilities are equal for all seven regions.

Question 80

On the most recent tax cut proposal, a random sample of Democrats and Republicans in the Congress cast their votes as follows.
Determine the expected frequencies for both the Democrats and Republicans who favor the tax cut proposal for the chi-square test of independence.