Question 1

A chi-squared distribution is symmetric.

Accepted Answer

A chi-squared distribution is not symmetric as it is skewed to the right.

Question 2

A left-tailed area in the chi-squared distribution equals .90.For 10 degrees of freedom the table value equals 15.9871.

Accepted Answer

The chi-squared distribution is a right-tailed distribution, so a left-tailed area of .90 corresponds to a right-tailed area of 1 - .90 = .10. The table value for a right-tailed area of .10 with 10 degrees of freedom is 15.9871.

Question 3

You cannot use a chi-squared goodness-of-fit test when there are only two possible outcomes for each trial in your experiment.

Accepted Answer

A chi-squared goodness-of-fit test can be used even when there are only two possible outcomes (e.g., success or failure) for each trial in an experiment; it is not limited by the number of categories or outcomes.

Question 4

A chi-squared test is used to describe a population of nominal data.

Accepted Answer

A chi-squared test is used to analyze and compare the expected and observed frequencies of categorical data, which includes nominal data.

Question 5

A left-tailed area in the chi-squared distribution equals .10.For 5 degrees of freedom the table value equals 9.23635.

Accepted Answer

This statement is false.

For a left-tailed area of .10 and 5 degrees of freedom in the chi-squared distribution, the table value should be 9.23635 or greater. However, the value given in the statement is already 9.23635, which means that the left-tailed area is actually greater than .10.

Question 6

All of the expected frequencies in a chi-squared goodness-of-fit test must be equal to each other.

Accepted Answer

In a chi-squared goodness-of-fit test, the expected frequencies are not required to be equal to each other; they are determined based on the null hypothesis and the sample data.

Question 7

A chi-squared goodness-of-fit test is always a two-tailed test.

Accepted Answer

A chi-squared goodness-of-fit test can be either one-tailed or two-tailed, depending on the research question and the hypotheses being tested. A one-tailed test is used when the researcher has a specific directional prediction (e.g., the observed frequencies will be significantly higher than expected), while a two-tailed test is used when the researcher does not have a specific directional prediction (e.g., the observed frequencies will be significantly different from expected).

Question 8

In conducting a chi-squared goodness-of-fit test,an essential condition is that all expected frequencies are at least five.

Accepted Answer

This condition is necessary to ensure the accuracy of the test statistics and the reliability of the p-value. If any expected frequency is less than five, the test may not be appropriate and alternative methods should be used.

Question 9

A small chi-squared test statistic in a goodness-of-fit test supports the null hypothesis.

Accepted Answer

A small chi-squared test statistic indicates that the observed data is similar to the expected data under the null hypothesis, supporting the idea that no significant difference exists between observed and expected data. Therefore, it supports the null hypothesis.

Question 10

For a chi-squared distributed random variable with 12 degrees of freedom and a level of significance of .05,the test statistics is 25.168.The chi-squared value from the table is 21.0261.These results will lead us to reject the null hypothesis.

Accepted Answer

The calculated test statistic (25.168) is greater than the critical value (21.0261) at a significance level of .05 with 12 degrees of freedom. Therefore, the results lead us to reject the null hypothesis.

Question 11

If the observed frequencies are all smaller than the expected frequencies in a goodness-of-fit test,the chi-squared test statistic will be negative.

Accepted Answer

The chi-squared test statistic cannot be negative, as it is the sum of squared differences between observed and expected frequencies.

Question 12

A multinomial experiment with two categories is identical to a binomial experiment.

Accepted Answer

A multinomial experiment with two categories can be thought of as a series of independent binomial experiments, one for each category. Therefore, it is identical to a binomial experiment.

Question 13

The only way the chi-squared test statistic can be zero is if the observed frequencies are all exactly the same as the expected frequencies.

Accepted Answer

This statement is true. The chi-squared test statistic is calculated by comparing the observed frequencies with the expected frequencies, and a difference between the two will result in a nonzero test statistic. If the observed frequencies are exactly the same as the expected frequencies, there will be no difference and the test statistic will be zero.

Question 14

The alternative hypothesis in a goodness-of-fit test is that none of the p_i values are equal to their values specified in H₀.

Accepted Answer

The alternative hypothesis in a goodness-of-fit test typically states that at least one of the observed frequencies does not match the expected frequency under the null hypothesis, not that none of the observed frequencies match.

Question 15

If the expected frequency of a cell is less than 5,you should combine cells of the table.

Accepted Answer

If the expected frequency of a cell is less than 5, it increases the chances of having a misleading result due to small sample size. Therefore, to avoid this, it is recommended to combine cells until the expected frequency in each cell is at least 5.

Question 16

The total of the observed frequencies in a multinomial experiment must equal nk where n is the number of trials and k is the number of categories.

Accepted Answer

The total of the observed frequencies in a multinomial experiment must equal n, the number of trials, not nk. Each trial results in one outcome from the k categories.

Question 17

In a goodness-of-fit test,all of the proportions specified in the null hypothesis must be equal to each other.

Accepted Answer

In a goodness-of-fit test, the proportions specified in the null hypothesis do not have to be equal to each other; they can be any set of proportions that are hypothesized for the categories being tested.

Question 18

If the expected frequency of a cell is less than 5,you should increase the significance level.

Accepted Answer

If the expected frequency of a cell is less than 5, it is generally recommended to use a different statistical test, such as Fisher's exact test, rather than increasing the significance level. Increasing the significance level may increase the likelihood of a type I error.

Question 19

If there are only two categories,the chi-squared goodness-of-fit test is the same as the z-test for p,the population proportion (as long as the sample/cell sizes meet the conditions).

Accepted Answer

When there are only two categories, the chi-squared goodness-of-fit test is equivalent to the z-test for the population proportion, as long as the sample/cell sizes meet the necessary conditions.

Question 20

For a chi-squared distributed random variable with 10 degrees of freedom and a level of significance of .025,the chi-squared table value is 20.4831.Suppose the value of your test statistic is 16.857.This will lead you to reject the null hypothesis.

Accepted Answer

The critical value from the chi-squared table at a level of significance of .025 and 10 degrees of freedom is 20.4831. Since the test statistic value of 16.857 is less than the critical value of 20.4831, we fail to reject the null hypothesis. Therefore, the answer is B.

Quiz 15: Chi-Squared Tests