Question 1

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

Accepted Answer

The z-test of a population proportion is used to test hypotheses about a population proportion, which is suitable for qualitative data with two categories.

Question 2

In testing the difference between two proportions using the normal distribution,we may use either a one-tailed or two-tailed test.

Accepted Answer

This is true. A one-tailed test is used when we have a prior expectation of the direction of the difference between the two proportions (e.g. we expect one proportion to be larger than the other), while a two-tailed test is used when we do not have a prior expectation of the direction of the difference.

Question 3

To describe a population with more than two categories you can only use a chi-squared goodness-of-fit test.

Accepted Answer

The answer of To describe a population with more than...

Question 4

If we want to perform a two-tail test for differences between two populations of nominal data with exactly two categories,we can employ either the z-test of p₁ − p₂,or the chi-squared test of a contingency table.

Accepted Answer

Both z-test and chi-squared test can be used to perform a two-tail test for differences between two populations of nominal data with exactly two categories.

Question 5

If we use the chi-squared method of analysis we must first check that there are at least 5 observations in each cell of the contingency table.

Accepted Answer

The answer of If we use the chi-squared method of...

Question 6

If we want to perform a one-tail test for differences between two populations of nominal data with exactly two categories,we must employ the z-test of p₁ − p₂.

Accepted Answer

The z-test of p1 - p2 is used to test for differences between two populations of nominal data with exactly two categories.

Question 7

The chi-squared distribution is used in:

Accepted Answer

The chi-squared distribution is used in all three scenarios: a goodness-of-fit test (testing whether a sample comes from a specific distribution), a test of a contingency table (testing for independence between two categorical variables), and describing a population having more than two categories. Therefore, choice D is the correct answer.

Question 8

If we want to conduct a two-tail test of a population proportion,we can employ:

Accepted Answer

A z-test of a population proportion is appropriate for testing hypotheses about a population proportion when the sample size is large enough to meet the necessary assumptions. The chi-squared test of a binomial experiment can also be used for this purpose, as it can be shown that the square of a z-test statistic follows a chi-squared distribution with one degree of freedom, making both A and B correct. The chi-squared test of a contingency table is not directly used for testing a single population proportion, making C incorrect. Therefore, both A and B are correct, leading to D as the correct answer.

Question 9

When we test for differences between two populations of nominal data with two categories,we can use only one technique,namely,the chi-squared test of a contingency table.

Accepted Answer

Other techniques, such as Fisher's exact test, can also be used for analyzing differences between two populations of nominal data with two categories, especially when sample sizes are small.

Question 10

Which statistical technique is appropriate when we compare two or more populations of qualitative data with two or more categories?

Accepted Answer

The chi-squared test of a contingency table is appropriate for comparing two or more populations of qualitative data with two or more categories. This test determines whether there is a significant association between the variables being compared. The other options, such as the z-test, are appropriate for comparing two populations with binary (yes/no) data, and the chi-squared goodness-of-fit test is appropriate for testing whether an observed distribution fits an expected distribution.

Question 11

Which of the following tests is appropriate for nominal data if the problem objective is to compare two populations and the number of categories exceeds 2?

Accepted Answer

The chi-squared test of a contingency table is appropriate for nominal data with more than 2 categories and when comparing two populations. Options A and B are appropriate for nominal data with 2 categories, while option D is too broad and includes options that may not be suitable for nominal data with multiple categories. Therefore, the best choice is option C.

Question 12

The squared difference between the observed and expected frequencies should be large if there is a significant difference between the proportions.

Accepted Answer

This is correct. The chi-square test statistic for testing differences between proportions is calculated by summing the squared differences between the observed and expected frequencies, divided by the expected frequencies. If there is a significant difference between the proportions, the observed frequencies will deviate from what would be expected based on chance alone, resulting in a large chi-square statistic.

Question 13

A test for the differences between two proportions can be performed using the chi-squared distribution.

Accepted Answer

The chi-squared test for independence can be used to compare proportions of two independent groups.

Question 14

If we want to perform a one-tail test of a population proportion p,we can employ either the z-test of p,or the chi-squared goodness-of-fit test.

Accepted Answer

The z-test is appropriate for testing a population proportion, while the chi-squared goodness-of-fit test is used for categorical data to see if observed frequencies match expected frequencies. They are not interchangeable for testing a single population proportion.

Question 15

If we want to conduct a one-tail test of a population proportion,we can employ:

Accepted Answer

The z-test of a population proportion is appropriate for a one-tailed test of a population proportion. The chi-squared test of a binomial experiment and the chi-squared test of a contingency table are not applicable for testing proportions.

Question 16

A test for whether one proportion is higher than the other can be performed using the chi-squared distribution.

Accepted Answer

A chi-squared distribution is used to test for independence between two categorical variables, not for comparing proportions. The appropriate test for comparing proportions is a z-test or a t-test.

Question 17

If we want to perform a two-tail test of a population proportion p,we can only use the z-test of p.

Accepted Answer

We can also use the chi-square test or the t-test for the two-tail test of a population proportion. The choice of test depends on the sample size and other factors.

Question 18

When we describe a population of nominal data,with exactly two categories,the multinomial experiment is actually a binomial experiment with one of the categorical outcomes labeled "success" and the other labeled "failure".

Accepted Answer

This statement is true. A multinomial experiment with two categories can be considered a binomial experiment with one category as "success" and the other as "failure".

Question 19

Which of the following tests does not use the chi-squared distribution?

Accepted Answer

The one tailed test for two proportions uses the normal distribution instead of the chi-squared distribution. The other options, test of a contingency table and goodness-of-fit test, both use the chi-squared distribution. Therefore, the best choice is C.

Question 20

Mathematical statisticians have established that if we square the value of z,the test statistic for the test of one proportion p,we produce the χ² statistic.That is,z² = χ².

Accepted Answer

This statement is true, as it is a well-established result in mathematical statistics. The test statistic for testing one proportion p is given by z = (p̂ - p)/√[p(1-p)/n], where p̂ is the sample proportion, p is the hypothesized proportion, and n is the sample size. Squaring both sides of this equation gives z^2 = [(p̂ - p)/√(p(1-p)/n)]^2 = (p̂ - p)^2/[p(1-p)/n]. This is equivalent to the χ^2 statistic with 1 degree of freedom, which is denoted as χ^2(1) or χ^2_1. Therefore, we have z^2 = χ^2_1.

Quiz 15: Chi-Squared Tests