Question 1

Consider a multinomial experiment with 200 trials, and the outcome of each trial can be classified into one of 5 categories. The number of degrees of freedom associated with the chi-square goodness-of-fit test equals:
A) 195
B) 40
C) 5
D) 4
E) 199

Accepted Answer

For a chi-square goodness-of-fit test with k categories and n independent trials, the degrees of freedom is (k-1). In this case, there are 5 categories, so k=5. Therefore, the degrees of freedom is (5-1) = 4.

Question 2

Historically, a bank has employed 50% permanent full-time workers, 25% permanent part-time workers, 15% temporary full-time workers, and 10% temporary part-time workers. In a recent sample of 400 workers, 190 were permanent full-time, 125 were permanent part-time, 55 were temporary full-time, and 30 were temporary part-time.  Use these data and  = 0.05 to assess whether there has been a change from the historical percentages.
Find  :
______________
This shows that one should ______________  .
Conclude that ______________ from the historical expectation.

Accepted Answer

The answer of Historically, a bank has employed 50% permanent...

Question 3

Which of the following is not a characteristic of a multinomial experiment?
A) The experiment consists of a fixed number, n, of trials.
B) The outcome of each trial can be classified into one of two categories called successes and failure.
C) The probabilitythat the outcome will fall into cell i remain constant for each trial.
D) Each trial of the experiment is independent of the other trials.
E) The experiment consists of a fixed number, n, of trials and the outcome of each trial can be classified into one of two categories called successes and failure.

Accepted Answer

A multinomial experiment involves more than two possible outcomes or categories, so option B which states that the outcome can only be classified as success or failure is not a characteristic of a multinomial experiment.

Question 4

Of the values for a chi-square test statistic listed below, which one is likely to lead to rejecting the null hypothesis in a goodness-of-fit test?
A) 0
B) 1.3
C) 1.9
D) 40
E) 1.5

Accepted Answer

A larger value for the chi-square test statistic indicates a greater discrepancy between the observed and expected frequencies, which would provide evidence against the null hypothesis in a goodness-of-fit test. A chi-square value of 40 is much larger than the other options listed and would be more likely to lead to rejecting the null hypothesis.

Question 5

If the expected frequency  for any cell i is less than 5, we:
A) must choose another sample of five or more observations
B) should use the normal distribution instead of the chi-square distribution
C) should combine the cells such that each observed frequencyis 5 or more
D) increase the number of degrees of freedom for the test by 5
E) all of these

Accepted Answer

If the expected frequency for any cell is less than 5, it can lead to unreliable results and violates the assumption of the chi-square test. Therefore, we need to combine the cells such that each observed frequency is 5 or more, which increases the expected frequency and reduces the risk of type-II error. Choosing another sample or using the normal distribution instead of the chi-square distribution does not address the issue of low expected frequency, and increasing the number of degrees of freedom could even worsen the problem.

Question 6

Which of the following statements is false?
A) The chi-square distribution is symmetrical.
B) The chi-square distribution is skewed to the right.
C) All values of the chi-square distribution are positive.
D) The critical region for a goodness-of-fit test with k categories is, whereis the value of the test statistic.
E) The chi-square distribution is symmetrical and all values of the chi-square distribution are positive.

Accepted Answer

The answer of Which of the following statements is false?
A)...

Question 7

In a goodness-of-fit test, suppose that the value of the test statistic is 13.08 and the number of degrees of freedom is 6. At the 5% significance level, the null hypothesis is:
A) rejected and p-value for the test is smaller than 0.05
B) not rejected, and p-value for the test is greater than 0.05
C) rejected, and p-value for the test is greater than 0.05
D) not rejected, and p-value for the test is smaller than 0.05
E) none of these

Accepted Answer

In a goodness-of-fit test, the null hypothesis is that the observed frequency distribution follows the expected theoretical distribution. A small p-value indicates that it is unlikely to get such a good match between the observed and expected values by chance, thus providing evidence to reject the null hypothesis.

In this case, since the test statistic is greater than the critical value (which is 12.592 according to a chi-square distribution table with 6 degrees of freedom and a 5% level of significance), the null hypothesis is rejected. Additionally, since the p-value for the test is smaller than 0.05 (the level of significance), this provides further evidence to reject the null hypothesis. Therefore, the answer is A.

Question 8

In the chi-square goodness-of-fit test, if the expected frequencies  and the observed frequencies  were quite different, we would conclude that the:
A) null hypothesis is false, and we would reject it
B) null hypothesis is true, and we would not reject it
C) alternative hypothesis is false, and we would reject it
D) chi-square distribution is invalid, and we would use the t-distribution instead

Accepted Answer

If the expected and observed frequencies are quite different, it suggests that the null hypothesis (which states that the observed frequencies are not significantly different from the expected frequencies) may not be true. Therefore, we would reject the null hypothesis and accept the alternative hypothesis that the observed frequencies are significantly different from the expected frequencies.

Question 9

A librarian knows from old records that the percentages of checked-out materials which are fiction, nonfiction, children's books, books on tape, and magazines are 35%, 25%, 20%, 10%, and 10%, respectively. The material checked out in each of the categories in a recent survey of 600 randomly chosen library items is listed below.  Use these data to assess whether there has been a change in the distribution of checked-out library materials. State the hypotheses to be tested and use  = 0.01 to interpret your results.
The hypotheses are:  :  = _______,  = _______,  = _______,  = _______,  = _______  :
______________ from the specified value(s).
Compute  :
______________
Thus we ______________  .
Conclude that ______________ from those specified in  .

Accepted Answer

The answer of A librarian knows from old records that...

Question 10

A casino customer is concerned about whether the dice the casino are using are "fair" (i.e., each face is equally likely to appear). The customer obtains a die that is being used by the casino and rolls it 240 times. The customer records the following data:  Compute the value of the test statistic.  = ______________
Set up the appropriate rejection region for  = 0.10.
Reject  when  > ______________.
What is the appropriate conclusion?
We ______________ conclude that the die is fair.

Accepted Answer

The answer of A casino customer is concerned about whether...

Question 11

Which statistical technique is appropriate when we describe a single population of qualitative data with two or more categories?
A) The z-test of the difference between two proportions.
B) The chi-square goodness-of-fit test.
C) The chi-square test of a contingency table.
D) The z-test of the difference between two proportions and the chi-square goodness-of-fit test.
E) The z-test of the difference between two proportions and the chi-square test of a contingency table.

Accepted Answer

The chi-square goodness-of-fit test is used to determine if a sample data matches a population. For qualitative data with multiple categories within a single population, this test is appropriate to see if the observed frequencies differ from the expected frequencies under a specific hypothesis.

Question 12

If we wish to determine whether there is evidence that the proportion of successes is higher in group 1 than in group 2, the appropriate test to use is the:
A) z test
B) test
C) t-test with 2 degrees of freedom
D) F-test with 2 degrees of freedom
E) all of these

Accepted Answer

The appropriate test for comparing the proportions of two independent groups is the z-test. The z-test is used when the sample size is large enough that we can assume a normal distribution for the sampling distribution of the proportion.

Question 13

If each element in a population is classified into one and only one of several categories, the population is:
A) a normal population
B) a multinomial population
C) a chi-square population
D) a binomial population
E) all of these

Accepted Answer

A population that is classified into one and only one of several categories is known as a multinomial population. A normal population refers to a continuous population with a normal distribution, while a chi-square population refers to a population with a chi-square distribution. A binomial population refers to a population with only two possible outcomes, which is not the case in this scenario. Therefore, the best choice is B.

Question 14

The sampling distribution of the test statistic for a goodness-of-fit test with k categories is:
A) Student t distribution with k-1 degrees of freedom
B) normal distribution
C) chi-square distribution with k-1 degrees of freedom
D) approximately chi-square distribution with k-1 degrees of freedom
E) approximately chi-square distribution with k degrees of freedom

Accepted Answer

The test statistic for a goodness-of-fit test follows an approximately chi-square distribution with k-1 degrees of freedom, where k is the number of categories.

Question 15

If we wish to determine whether there is evidence that the proportion of successes is the same in group 1 as in group 2, the appropriate test to use is:
A) the z test
B) thetest
C) both the z test and thetest
D) neither the z test nor thetest

Accepted Answer

The z test is used to compare the proportions of successes in two independent groups, while the test is used to compare the proportions of successes in two dependent groups. In this case, we are comparing the proportions of successes in two independent groups, so the appropriate test to use is the z test.

Question 16

A sample survey of 850 people in Chicago was conducted in 2001 to see how people get to work. The results are shown in the frequency table below.  A study of the same nature was conducted in 1990 and revealed 12% walk, 67% drive an automobile, 18% take a bus, and 3% ride a bicycle.
Do the current data support that the proportions are the same as in 1990? Use  .
Compute  :
______________
One should ______________  .
Conclude that the current data ______________ that the proportions in 2001 are the same as in 1990.

Accepted Answer

The answer of A sample survey of 850 people in...

Question 17

Which of the following statements is false?
A) The rejection region of the chi-square goodness-of-fit test has k-1 degrees of freedom, where k is the number of categories (called cells).
B) The rejection region of the chi-square goodness-of-fit test is, where k is the number of categories andis the value of the test statistics.
C) The chi-square goodness-of-fit test involves two categorical variables.
D) The chi-square distribution is skewed to the right.
E) None of these.

Accepted Answer

The chi-square goodness-of-fit test involves only one categorical variable, which is being compared to an expected distribution. Therefore, statement C is false.

Question 18

To determine the critical values in the chi-square distribution table, the process requires the following:
A) degrees of freedom
B) probability of Type I error
C) probability of Type II error
D) degrees of freedom and probability of Type I error
E) degrees of freedom and probability of Type II error

Accepted Answer

The critical values in the chi-square distribution table are determined based on the degrees of freedom and probability of Type I error (also known as the significance level). Therefore, the correct choice is D, which involves both of these variables. The probability of Type II error is not used in determining critical values in the chi-square distribution table.

Question 19

A national survey states that 67% of college students are under the age of 25, 21% are between the ages of 25 and 30, 8% are between 30 and 40, and 4% are over 40. A random sample of 250 students at Grand Rapids Community College yielded the following data:  Test whether the distribution of students' ages at Grand Rapids Community College agrees with the national survey.
Compute the value of the test statistic.
Compute  .
______________
Set up the appropriate rejection region for  = 0.05.
Reject  when  > ______________.
What is the appropriate conclusion?
______________  .
Conclude the distribution of students' ages at Grand Rapids Community College ______________ agree with the national survey.

Accepted Answer

The answer of A national survey states that 67% of...

Question 20

A total of 200 people in each of three grocery stores were asked if they favored, opposed, or were indifferent to the sale of lottery tickets in grocery stores.  The results of the survey are summarized below. Use the MINITAB computer output shown here to conduct a chi-square test of independence at  = 0.05.
The data show that one should ______________  .
Thus we ______________ conclude that the two variables are dependent; that is, the opinion of people is not the same at the three stores.

Accepted Answer

The answer of A total of 200 people in each...

Quiz 14: Analysis of Categorical Data

Consider a multinomial experiment with 200 trials, and the outcome of each trial can be classified into one of 5 categories. The number of degrees of freedom associated with the chi-square goodness-of-fit test equals:

Which of the following is not a characteristic of a multinomial experiment?

Of the values for a chi-square test statistic listed below, which one is likely to lead to rejecting the null hypothesis in a goodness-of-fit test?

If the expected frequency for any cell i is less than 5, we:

Which of the following statements is false?

In a goodness-of-fit test, suppose that the value of the test statistic is 13.08 and the number of degrees of freedom is 6. At the 5% significance level, the null hypothesis is:

In the chi-square goodness-of-fit test, if the expected frequencies and the observed frequencies were quite different, we would conclude that the:

Which statistical technique is appropriate when we describe a single population of qualitative data with two or more categories?

If we wish to determine whether there is evidence that the proportion of successes is higher in group 1 than in group 2, the appropriate test to use is the:

If each element in a population is classified into one and only one of several categories, the population is:

The sampling distribution of the test statistic for a goodness-of-fit test with k categories is:

If we wish to determine whether there is evidence that the proportion of successes is the same in group 1 as in group 2, the appropriate test to use is:

Which of the following statements is false?

To determine the critical values in the chi-square distribution table, the process requires the following: