2 the manager of a supermarket chain wants to determine...

2. The manager of a supermarket chain wants to determine whether product location has any effect on the sale of pet toys. Three different aisle locations are considered: front, middle, and rear. A random sample of 18 stores is selected with 6 stores randomly assigned to each aisle location. The size of the display area and price of the product are constant for all stores. At the end of a 1-month trial period, the sales volumes (in thousands of dollars) of the product in each store were as follows: Aisle Location Front Middle Rear 8.8 4.0 4.4 7.5 2.4 5.0 5.4 2.4 3.8 6.2 1.8 2.6 5.0 2.2 2.0 4.0 2.0 2.6 a. At the 0.05 level of significance, is there evidence of a difference in the mean sales among the various aisle locations ? Determine ANOVA single factor results using Excel to answer this question. b. What is your statistical conclusion for this problem?

Transcript

00:01 Once again, welcome to a new problem.

00:05 This time we're dealing with hypothesis testing.

00:08 We're dealing with hypothesis testing.

00:13 And when it comes to hypothesis testing, we have the anova.

00:18 And the anova stands for the analysis of variance.

00:26 So the anova stands for the analysis of variance.

00:29 And it compares means of at least two or more independent groups, at least two or more independent groups, at least two or more independent groups for significance.

00:57 So that's the anova.

00:59 And the test statistic for the anova is the same as the f ratio and the f ratio is such that we have ms between all of our ms error so this is the mean square between groups and we also have the mean square between groups and we also have the mean square between groups and we also have the mean square square error.

01:38 We have the main square between groups and we're also looking at the main square for the error.

01:47 We do have the requirements for the different groups.

01:57 We have the requirements for different groups.

01:59 For example, we have sample size so this is your typical and over table and then we have the the sample mean and then of course we have the sample standard deviation, sample standard deviation.

02:21 And for instance, if we do have three groups, then what's going to happen is that we can have a group one and then we could also have a group two and we could have information for group two and we could have information for group 3.

02:40 So when it comes to the sample sizes, this is n1, n2, and n3.

02:48 And for the means, when it comes to the sample means, this is x -bar 1, x -bore 2, and x -bar 3.

02:57 Remember, your typical x -bar sample mean is taking the summation of all the data and then divided by the sample size, which is n.

03:09 And in terms of the standard deviation, we have the standard deviation for the first group, for the second group, and then also for the third group.

03:19 Remember the f ratio, as we talked about it, is the same as nj, x by j minus x bar squared, all over k minus 1, all over sum of x minus x minus x, x by j squared on over n minus k and there are certain requirements on this one remember we're using the f distribution we're using the f distribution that's your main interest and the degrees of freedom for the top is k minus 1 this is the numerator remember k stands for the number of groups.

04:13 So if we have three groups you could see that the degrees of freedom is 3 minus 1.

04:19 And then in the denominator we also have n minus k where n is the total sample size for the three groups.

04:31 And k is the number of groups.

04:36 So k stands for the number of groups that you're looking at.

04:41 N sub j is the same as either n1 or n2 or n3 up until n j where j is the group specific group that we're dealing with in the problem and x sub j is either x1 or x2 or x2 x bar 2 which is the mean the means of the specific groups up until the mean of the j group.

05:17 The test statistic, the test statistic, which represents the anova table, would be f test.

05:27 As long as the f test is greater than the f -quidical, the test is significant, so we reject the nar hypothesis.

05:37 So you're going to reject the nar hypothesis.

05:39 This is also in the table, we have another table that produces the results of the anova.

05:49 So this is the results table and we have a source of variation as the first column.

05:56 And then we also have the sum of squares.

06:00 So we have the sum of squares.

06:02 We have the degrees of freedom.

06:05 We have the mean square and of course we have the f ratio so the source of variation can either be between groups or between treatments and so this is ssb sum of squares between groups which is sum of n subj x bar we actually we need a little bit of space right here.

06:40 So we have some of, so we're looking for source of variation.

06:47 Looking for source of variation and this is obviously going to be the source of variation is going to be some of squares for between groups or between treatments which is n subj.

07:07 X bar j minus x bar squared the degrees of freedom would be k minus one that's between the groups for example if you have three groups that's 3 minus 1 and then some of mean square between groups becomes some of squares between groups of the degrees of freedom which is k minus 1 and then of course we have the other portion and the second portion is the error and the sum of squares for the error becomes the same as summation of x minus x by j.

07:51 Remember x is the data value, any specific data value that you're dealing with, and then n minus k.

08:01 And then the mean square error becomes the sum of squares of error over the degrees of freedom for error, which is n minus k.

08:12 And of course, we do have other options.

08:17 So this is going to be an extension like that.

08:21 We have a total...

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