Question 1

An F-statistic is ________.

Accepted Answer

An F-statistic is the ratio of two variances. It is typically used in analysis of variance (ANOVA) to test the hypothesis that the means of several groups are equal.

Question 2

One characteristic of the F-distribution is that the computed F can only range between -1 and +1.

Accepted Answer

This statement is false. The F-distribution can take on values from 0 to positive infinity. It cannot be negative.

Question 3

In an ANOVA table,k represents the total number of sample observations and n represents the total number of treatments.

Accepted Answer

In an ANOVA table, k typically represents the number of groups or levels within a treatment, while n represents the sample size within each group or level. So, k and n have specific meanings and are not interchangeable.

Question 4

A large department store examined a sample of 18 credit card sales and recorded the amounts charged for each of three types of credit cards: MasterCard,Visa,and Discover. Six MasterCard sales,seven Visa,and five Discover sales were recorded. The store used an ANOVA to test if the mean sales for each credit card were equal. What are the degrees of freedom for the F-statistic?

Accepted Answer

The degrees of freedom for the F-statistic in an ANOVA are calculated as follows: the numerator degrees of freedom are one less than the number of groups (k - 1), and the denominator degrees of freedom are the total number of observations minus the number of groups (N - k). Here, k = 3 (MasterCard, Visa, Discover) and N = 18 (total sales). Thus, numerator degrees of freedom = 3 - 1 = 2, and denominator degrees of freedom = 18 - 3 = 15.

Question 5

When a blocking effect is included in an ANOVA,the analysis is more likely to detect differences in the treatment means.

Accepted Answer

Including a blocking effect in an ANOVA can help to reduce the variability in the data and better detect treatment differences, making it more likely to find significant differences among the treatment means.

Question 6

What distribution does the F-distribution approach as the sample size increases?

Accepted Answer

As the sample size increases, the F-distribution approaches a Normal distribution because of the central limit theorem.

Question 7

If the computed value of F is 0.99 and the F critical value is 3.89,we would not reject the null hypothesis.

Accepted Answer

The computed F value of 0.99 is less than the F critical value of 3.89. Therefore, we would not reject the null hypothesis.

Question 8

If a confidence interval for the difference between a pair of treatment means includes 0,then we reject the null hypothesis that there is no difference in the pair of treatment means.

Accepted Answer

If the confidence interval for the difference between a pair of treatment means includes 0, it means that the difference between the means is not statistically significant at the given level of confidence. This does not necessarily mean that we reject the null hypothesis. The decision to reject or fail to reject the null hypothesis depends on the level of significance and the p-value calculated from the t-test or other statistical methods.

Question 9

Interaction between two factors occurs when the effect of one factor on the response variable is the same for any value of another factor.

Accepted Answer

Interaction between two factors occurs when the effect of one factor on the response variable depends on the level of another factor. Therefore, the statement "the effect of one factor on the response variable is the same for any value of another factor" is false.

Question 10

The F-distribution's curve is positively skewed.

Accepted Answer

The F-distribution is positively skewed because it is bounded below by 0 (it cannot take negative values) and has a long right tail extending towards positive infinity.

Question 11

For an ANOVA test,rejecting the null hypothesis does not identify which treatment means differ significantly.

Accepted Answer

Rejecting the null hypothesis in an ANOVA test only indicates that there is a significant difference among at least two treatment means, but it does not specify which ones. Further analysis, such as post-hoc tests or pairwise comparisons, is needed to determine which treatment means differ significantly from each other.

Question 12

When a blocking effect is included in an ANOVA,the result is a smaller error sum of squares.

Accepted Answer

When a blocking effect is included in an ANOVA, the variability attributed to the blocking factor is removed from the error sum of squares, resulting in a smaller error sum of squares. This increases the power of the analysis to detect any significant differences between treatments of interest.

Question 13

Which statement is correct about the F-distribution?

Accepted Answer

The F-distribution cannot be negative as it is always a positive number. Options B, C, and D are incorrect because the F-distribution is not the same as the t-distribution or z-distribution.

Question 14

If we want to determine which treatment means differ,we compute a confidence interval for the difference between each pair of means.

Accepted Answer

This statement is true. Confidence intervals can be used to determine if two means are statistically different by checking if the interval contains zero (indicating no difference) or if it does not contain zero (indicating a significant difference).

Question 15

In a two-way ANOVA with interaction,there are two factor effects and an interaction effect.

Accepted Answer

In a two-way ANOVA with interaction, there are two main effects (factor effects) and an interaction effect. The factor effects refer to the effects of each individual factor (independent variable) on the dependent variable, while the interaction effect refers to the combined effect of the two factors on the dependent variable.

Question 16

To employ Analysis of Variance (ANOVA),the populations being studied must be approximately normally distributed.

Accepted Answer

One of the assumptions of ANOVA is that the populations being compared are approximately normally distributed. This means that the data within each group being compared should follow a normal distribution. If this assumption is not met, the results of ANOVA may not be reliable or accurate.

Question 17

For the hypothesis test,H₀: σ₁²= σ₂²,with n₁ = 10 and n₂ = 10,the F-test statistic is 2.56. At the 0.02 level of significance,we would reject the null hypothesis.

Accepted Answer

To determine whether to reject the null hypothesis, we need to compare the F-test statistic to the critical value from the F-distribution table at the given level of significance for the degrees of freedom associated with the sample sizes. Without knowing the critical value for the specific degrees of freedom and the exact level of significance (two-tailed or one-tailed), we cannot definitively say the null hypothesis would be rejected based solely on the F-test statistic value.

Question 18

Analysis of variance is used to ________.

Accepted Answer

Analysis of variance (ANOVA) is a statistical method used to determine whether there is a significant difference between the means of two or more groups. It is used to simultaneously compare several population means, making option D the correct choice. Option A is incorrect because nominal data is compared using chi-square analysis. Option B is too vague and can refer to several different types of tests. Option C is incorrect because comparing population proportions is done using tests like the z-test or chi-square test of independence.

Question 19

The alternative hypothesis used in ANOVA is H₁: All population means are equal.

Accepted Answer

The alternative hypothesis used in ANOVA is typically the opposite of the null hypothesis, which is that at least one population mean is different from the others. Therefore, the correct alternative hypothesis for ANOVA is HS1U1B11S1U1B1: At least one population mean is different from the others.

Question 20

To employ ANOVA,the populations should have approximately equal standard deviations.

Accepted Answer

It is a requirement of ANOVA that the populations being compared have approximately equal standard deviations. This is necessary in order for the test to be valid and accurate. If the standard deviations are drastically different, it can lead to biased results and make it difficult to interpret the differences between the groups being compared.

Quiz 12: Analysis of Variance