ANOVA

Question 1

In words and symbols, what is the null hypothesis in one-way ANOVA? In words and symbols, what is the alternative hypothesis in one-way ANOVA?

Answer 1

H0: µ1 = µ2 = . . . = µk (The population means of the k groups are all equal.) Ha: µi does not equal µj for at least one i, j pair. (The population means of the k groups are not all equal.)

Question 2

is sample means are very different then SST is

Answer 2

large, if sample means similar SST is small *looking at variability between groups

Question 3

What is SSE

Answer 3

variability WITHIN groups,

Question 4

what is mean square

Answer 4

sum of squares/dof *Sp^2=MSE

Question 5

what is the F statistic

Answer 5

ratio of mean square treatment/mean square error (last two columns)

Question 6

what happens of H0 is false

Answer 6

MST tends to be bigger than MSE and the test statistic tends to be large *greater value of F statistic the greater evidence against null

Question 7

how do u find Sp^2

Answer 7

MSE

Question 8

determine number of pairwise comparisons

Answer 8

(# of groups/2) ** doing that n!/x!(n-x)! thing

Question 9

In a one-way ANOVA the F statistic is found to be 0.00002. Does this value give strong evidence against the null hypothesis?

Answer 9

No. In one-way ANOVA, only large values of the F test statistic give strong evidence against the null hypothesis.

Question 10

• Under what conditions would the test statistic be equal to 0? - Under what conditions would the test statistic be equal to 1?

Answer 10

• The test statistic will equal 0 if MST = 0 (provided MSE 6= 0). - MST will equal 0 when there is no variability in the sample means—when the k groups all have the same value of the sample mean

Question 11

. Consider the boxplots in Figure 1, representing 3 separate and independent samples of size 20. Which of the following statements are true?

(a) If we performed a one-way ANOVA on these 3 samples, then the p-value would be small.
(b) If we carried out a t-test of H0: µA = µC against a two-sided alternative, then the p-value would be small.
(c) The use of the ANOVA procedures for the test of H0: µA = µB = µC would be a bad idea, as the assumptions are clearly violated.
(d) If we carried out a pooled-variance t-test of H0: µA = µC against a two-sided alternative, and a one-way ANOVA F test of H0: µA = µC , then the test statistics would have the relationship: t 2 = F.
(e) If we carried out a pooled-variance t-test of H0: µA = µC against a two-sided alternative, then a one-way ANOVA of the test of H0: µA = µC , the p-values would be exactly equal.

Answer 11

(a) True. Visually there is very, very strong evidence against H0, implying a small p-value .
(b) True. Visually there is very, very strong evidence against H0, implying a small p-value .
(c) False. There is no visual evidence against the assumptions of normality and a common population variance.
(d) True. For two-sample problems the pooled-variance t procedure and one-way ANOVA are equivalent tests, with t 2 = F.
(e) True. For two-sample problems the pooled-variance t procedure and one-way ANOVA are equivalent tests, resulting in the exact same p-value.

Question 12

one-way ANOVA, with 10 observations in each of 5 different groups. Suppose the null hypothesis is true, and the assumptions are true.

(a) What is the distribution of the test statistic?
(b) What is the distribution of the p-value?
(c) What would the p-value equal on average?

Answer 12

(a) The test statistic will have an F distribution with 4 df in the numerator, and 45 df in the denominator. If the null hypothesis and the assumptions are true, the F test statistic has an F distribution with k − 1 degrees of freedom in the numerator and n − k degrees of freedom in the denominator. If there are 10 observations in each of 5 groups, there are 5 − 1 = 4 degrees of freedom in the numerator, and 50 − 5 = 45 degrees of freedom in the denominator.
(b) If the null hypothesis and the assumptions are true, then the p-value will have a uniform distribution between 0 and 1.
(c) Since the p-value is uniformly distributed between 0 and 1, the p-value will equal 0.5 on average (if the null hypothesis and the assumptions are true).

Question 13

(a) If the null hypothesis (and the assumptions) are true, then the test statistic in one-way ANOVA has an F distribution.

Answer 13

true

Question 14

(b) In one-way ANOVA, we assume that the observations within each group are normally distributed, and that all groups have the same population variance.

Answer 14

true

Question 15

(c) In one-way ANOVA, we assume that the observations within each group are normally distributed, and that all groups have the same population mean.

Answer 15

false

Question 16

(d) The test statistic in one-way ANOVA can be negative.

Answer 16

false, F test stat is a ratio of variances which can never be negative

Question 17

(e) If the null hypothesis is false, then MST will tend to be bigger than MSE.

Answer 17

true

Question 18

If the null hypothesis is true, then the F statistic will be infinite

Answer 18

false

Question 19

If the null hypothesis is true, then the F statistic will equal 1

Answer 19

False. If H0 is true, the F statistic will have a median that is near 1, but it can take on any value between 0 and ∞.

Question 20

If the null hypothesis is false, then the F statistic will sometimes be less than one

Answer 20

True. The F statistic can take on any nonnegative value, regardless of whether the null hypothesis is true or false

Question 21

If the null hypothesis is false, then the F statistic will sometimes be less than 0.

Answer 21

False. The F statistic cannot be negative.

Question 22

If the null hypothesis is false, then the p-value will be less than 0.05

Answer 22

False. If the null hypothesis is false, we can still end up with a large p-value.

Question 23

If MSE = 0, then the p-value will equal the F statistic.

Answer 23

false

Question 24

If the population means are equal, then the F statistic will equal 1.

Answer 24

False. If the population means are equal, then H0 is true and the F statistic will have an F distribution

Question 25

If the sample means are all equal, then the F statistic will equal 1.

Answer 25

False. If the sample means are equal, the F statistic will equal 0

Question 26

If the sample means are all equal, then the null hypothesis is true

Answer 26

False. If the sample means are all equal, then there is absolutely no evidence whatsoever against H0. But that still does not mean it must be true.

Question 27

The mean square error is the pooled variance.

Answer 27

True.

Question 28

If the population means are very different, we expect the F statistic to be greater than one.

Answer 28

True. If H0 is true (the population means are equal), then the F statistic will have a median that is somewhere around 1. If H0 is false, the F statistic will tend to be larger than 1.

Question 29

We will reject the null hypothesis at the 5% level whenever the sample means are all equal

Answer 29

False. When the sample means are all equal there is absolutely no evidence whatsoever against the null hypothesis

Question 30

An assumption of one-way ANOVA is that the groups all have different population variances.

Answer 30

False. One-way ANOVA assumes the population variances are equal.

Question 31

If the assumptions of the pooled-variance t procedures are met, then the t test and the F test are equivalent tests, with t 2 = F.

Answer 31

true

Question 32

The equal variance assumption is not important for large sample sizes, due to the central limit theorem.

Answer 32

. False. A violation of the equal variance assumption can be very problematic, even for large sample sizes. The central limit theorem can help out with a violation of the normality assumption.

Question 33

what is MSE and MST mean

Answer 33

variabily within groups, MST is variability of group means