Color affects us in many ways. For example, Exercise C.92 on...

Color affects us in many ways. For example, Exercise C.92 on page 498 describes an experiment showing that the color red appears to enhance men's attraction to women. Previous studies have also shown that athletes competing against an opponent wearing red perform worse, and students exposed to red before a test perform worse. $^{3}$ Another study $^{4}$ states that "red is hypothesized to impair performance on achievement tasks, because red is associated with the danger of failure." In the study, US college students were asked to solve 15 moderately difficult, five-letter, single-solution anagrams during a 5-minute period. Information about the study was given to participants in either red, green, or black ink just before they were given the anagrams. Participants were randomly assigned to a color group and did not know the purpose of the experiment, and all those coming in contact with the participants were blind to color group. The red group contained 19 participants and they correctly solved an average of 4.4 anagrams. The 27 participants in the green group correctly solved an average of 5.7 anagrams and the 25 participants in the black group correctly solved an average of 5.9 anagrams. Work through the details below to test if performance is different based on prior exposure to different colors. (a) State the hypotheses. (b) Use the fact that sum of squares for color groups is 27.7 and the total sum of squares is 84.7 to complete an ANOVA table and find the F-statistic. (c) Use the F-distribution to find the p-value. (d) Clearly state the conclusion of the test.

Color affects us in many ways. For example, Exercise C.92 on page 498 describes an experiment showing that the color red appears to enhance men's attraction to women. Previous studies have also shown that athletes competing against an opponent wearing red perform worse, and students exposed to red before a test perform worse. $^{3}$ Another study $^{4}$ states that "red is hypothesized to impair performance on achievement tasks, because red is associated with the danger of failure." In the study, US college students were asked to solve 15 moderately difficult, five-letter, single-solution anagrams during a 5-minute period. Information about the study was given to participants in either red, green, or black ink just before they were given the anagrams. Participants were randomly assigned to a color group and did not know the purpose of the experiment, and all those coming in contact with the participants were blind to color group. The red group contained 19 participants and they correctly solved an average of 4.4 anagrams. The 27 participants in the green group correctly solved an average of 5.7 anagrams and the 25 participants in the black group correctly solved an average of 5.9 anagrams. Work through the details below to test if performance is different based on prior exposure to different colors.
(a) State the hypotheses.
(b) Use the fact that sum of squares for color groups is 27.7 and the total sum of squares is 84.7 to complete an ANOVA table and find the F-statistic.
(c) Use the F-distribution to find the p-value.
(d) Clearly state the conclusion of the test.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence in a sample of data to infer that a certain condition is true for the entire population. It involves formulating a null hypothesis (typically stating that there is no effect or no difference) and an alternative hypothesis (indicating that there is an effect or a difference), then using data to determine which hypothesis is more consistent with the observed results.

Analysis of Variance (ANOVA)

ANOVA is a statistical technique used to compare the means of three or more groups to see if at least one group is statistically different from the others. It does this by analyzing the variances both within groups and between groups. The method is especially useful when testing for differences across multiple groups, rather than performing multiple pairwise tests.

F-Statistic

The F-statistic is a ratio used in ANOVA to compare the variance between group means to the variance within the groups. A larger F-statistic indicates that the between-group variability is large relative to the within-group variability, which may suggest that at least one group mean is significantly different from the others.

P-Value

The p-value in hypothesis testing measures the probability of obtaining test results at least as extreme as the ones observed, under the assumption that the null hypothesis is true. A low p-value indicates that such extreme results are unlikely under the null hypothesis, leading researchers to consider rejecting the null in favor of the alternative hypothesis.

Sum of Squares

Sum of squares is a measure used in statistics to quantify the total variability in a set of data. In the context of ANOVA, the total sum of squares is partitioned into the between-group sum of squares and the within-group sum of squares. This partitioning helps in determining how much of the overall variability is explained by the differences between group means.

Transcript

00:01 In this question, we have a randomized double -blind study that asked u .s.

00:05 College students to solve 15 anagrams in five minutes.

00:08 Just before they did this test, students received information about the study written either in red, green, or black ink.

00:16 So we are going to test whether performance, which is measured in the number of anagrams solved, is different based on ink color.

00:23 The first thing that we'll do is take a look at the table shown here below where we have.

00:30 Have the three groups, and it's important to note that we have three groups, red, green, and black.

00:37 And we're now looking at the size of each group, 19, 27, and 25.

00:44 That gives us a total of 71 students.

00:49 Let's go ahead and answer part a by writing the null and alternative hypotheses.

00:54 So here we're going to state that the means are equal.

01:00 So mu 1 equals, well, let's change that.

01:05 How about if we call it mu r for red? mu r equals mu g equals mu b for black.

01:15 So that is our null hypothesis that all three means are equal.

01:19 So then the alternative would be that some two means are not equal, and i'll write it this way.

01:25 Sum mu i is not equal to mu j all right let's go ahead and complete the anova table now the anova table was partially filled in for us and what we see is that the sum of squares for the groups is 27 .7 and the sum of squares for the total is 84 .7 that means that if we take the difference that will be the sum of squares for the error, which is within each group.

01:56 So that works out to be 57 when we do that subtraction.

02:00 The next thing that we can fill in is the degrees of freedom.

02:06 For the group, remember, the degrees of freedom is k minus one.

02:12 So here we have k, which was three groups.

02:15 So we have three minus one, which is two.

02:18 So three minus one is two degrees of freedom.

02:22 And then for the total, the number of degrees of freedom is n minus 1, and our total number of students was 71.

02:31 So here we have 71 minus 1 equals 70...