Question 1

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

Accepted Answer

This is true. When expected frequencies are less than five, the chi-squared test may not produce accurate results and other methods should be considered.

Question 2

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 is because the formula for the chi-squared test statistic involves subtracting the observed frequencies from the expected frequencies and squaring the result. If the observed frequencies are all exactly the same as the expected frequencies, then the difference will be zero for every category, and therefore the sum (and thus the chi-squared test statistic) will be zero as well.

Question 3

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

False

Question 4

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

Accepted Answer

A multinomial experiment with two categories is the same as a binomial experiment because there are only two possible outcomes for each trial. The only difference is that in a multinomial experiment, there are more than one trial with different possible outcomes, while in a binomial experiment, there is only one trial.

Question 5

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. It is always a non-negative value.

Question 6

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 7

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 8

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 hypothesis being tested.

Question 9

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

Accepted Answer

A chi-squared test is a statistical test used to determine whether there is a statistically significant association between two or more nominal variables within a population. Therefore, it is used to describe a population of nominal data.

Question 10

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 11

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 answer of A left-tailed area in the chi-squared distribution...

Question 12

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 13

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 14

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

Since the test statistic of 25.168 is greater than the critical value of 21.0261, we have sufficient evidence to reject the null hypothesis at a level of significance of .05. Therefore, the answer is True.

Question 15

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

This is true. In the case of only two categories, the chi-squared goodness-of-fit test can be replaced by the z-test for p, the population proportion. The conditions for the sample size and cell sizes remain the same as in the chi-squared test.

Question 16

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

Accepted Answer

When the expected frequency of a cell is less than 5, combining cells may be necessary in order to maintain the validity of the statistical test being performed. This is known as the "small expected frequency" rule.

Question 17

A chi-squared distribution is symmetric.

Accepted Answer

The chi-squared distribution is not symmetric. It is a right-skewed distribution, meaning that there is a long tail to the right side. This is because the values cannot be negative and tend to be quite large due to the squaring of the standard normal deviates.

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 recommended to use alternative statistical methods such as Fisher's exact test or to combine adjacent cells to increase the expected frequency. Increasing the significance level is not a valid solution as it does not address the issue of low expected frequency.

Question 19

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 does not deviate significantly from the expected values under the null hypothesis, providing evidence in support of it.

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

Quiz 15: Chi-Squared Tests