91Ó°ÊÓ

The largest manufacturer of cold breakfast cereal in Spokane, Washington notices that sales of its hottest selling cereal, Crispy Crunchy Sugar-Coated Corn Chips, are slipping. In a marketing research study, the company experiments with three new package designs in the hope of finding one that will reverse this trend. The company thinks that the area in which the cereal is marketed will have an effect on sales and thus divides the state into 4 geographical areas. Five nearly identical stores are selected in each area to be used for the experiment. After a month sales in each store are recorded and an analysis of variance table is constructed. No interactions between package design and geographical area are found. Thus the ANOVA table is $$ \begin{array}{|l|l|l|l|} \hline \begin{array}{l} \text { Source of } \\ \text { Variation } \end{array} & \begin{array}{l} \text { Sum of } \\ \text { Squares } \end{array} & \text { Degrees of Freedom } & \begin{array}{l} \text { Mean } \\ \text { Square } \end{array} \\ \hline \text { Package Design } & 154.24 & 3-1=2 & 77.12 \\ \hline \begin{array}{l} \text { Geographical } \\ \text { Area } \end{array} & 218.00 & 4-1=3 & 72.66 \\ \hline \text { Error } & 746.01 & 29-2-3=24 & 31.08 \\ \hline \end{array} $$ Test at the \(.025\) level, for significant differences in sales due to differences in package designs and geographical area.

Step by step solution

01

Identify the given information and required tests

We are given: - Significance level: \(0.025\) - Degrees of freedom: - Package Design: \(df_{1} = 2\) - Geographical Area: \(df_{2} = 3\) - Error: \(df_{3} = 24\) - Mean Squares: - Package Design: \(MS_{1} = 77.12\) - Geographical Area: \(MS_{2} = 72.66\) - Error: \(MS_{E} = 31.08\) We will be conducting two separate tests, one for package design and the other for geographical area.

02

Calculate F-values for Package Design and Geographical Area

To determine if there is a significant difference in sales due to package design and geographical area, we need to compute the F-values for both. The F-value is calculated by taking the appropriate mean square value and dividing it by the mean square error. For Package Design, F-value is: \(F_{1} = \frac{MS_{1}}{MS_{E}} =\frac{77.12}{31.08} = 2.48\) For Geographical Area, F-value is: \(F_{2} = \frac{MS_{2}}{MS_{E}} =\frac{72.66}{31.08} = 2.34\)

03

Obtain F-distribution critical values

Utilizing the F-distribution table at a 0.025 significance level and with the given degrees of freedom for Package Design and Geographical Area, we can obtain the F-distribution critical values. For Package Design, critical F-value: \(F_{critical1} (2, 24; 0.025)\) For Geographical Area, critical F-value: \(F_{critical2} (3, 24; 0.025)\) When looking up these values in the F-distribution table, we find: \(F_{critical1} = 5.76\) \(F_{critical2} = 4.71\)

04

Compare F-values against F-distribution critical values

Now, we will compare our calculated F-values with the critical F-values from the table. If the calculated F-value is greater than or equal to the critical F-value, then there is a significant difference in sales due to that factor. For Package Design: \(F_{1} = 2.48 < F_{critical1} = 5.76\) For Geographical Area: \(F_{2} = 2.34 < F_{critical2} = 4.71\)

05

Conclude the results

Since both F-values are less than their respective critical F-values, we cannot reject the null hypothesis. Therefore, we can conclude that there is no significant difference in sales due to differences in package designs and geographical areas at a 0.025 significance level.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

ANOVA

Analysis of Variance, commonly known as ANOVA, is a statistical method used to compare the means of three or more samples to find out if at least one sample mean is significantly different from the others. It's akin to multiple two-sample t-tests, but adjusted to prevent the inflation of type I errors that occurs with multiple comparisons.

The cereal company in our exercise is using ANOVA to determine whether package designs and geographical areas significantly impact sales. By comparing the sales means across different designs and areas, the company can assess if variations in these factors correlate with variations in sales. ANOVA works by analyzing the variation within groups (the natural fluctuation in sales for a given package design or area) and the variation between groups (how much the sales means differ from each other across designs or areas).

To improve the understanding of ANOVA results, it's essential to check the assumptions underlying the test, such as normality and homogeneity of variance, to ensure the reliability of the conclusions.

F-distribution

The F-distribution is the theoretical distribution that the test statistic of an ANOVA, called the F-value, follows under the null hypothesis (the assumption that there is no effect or difference). This distribution is right-skewed and depends on two sets of degrees of freedom: the degrees of freedom for the effect being tested (e.g., package design), and the degrees of freedom for the error.

The comparison between the F-values from the ANOVA and the critical F-values from the F-distribution table allows researchers to make decisions about the null hypothesis. If the calculated F-value is larger than the critical value, the null hypothesis is rejected, indicating that there are significant differences between group means. In our exercise, the test determined that neither package design nor geographical area had a significant impact on sales at the .025 significance level.

Degrees of Freedom

Degrees of freedom (df) are the number of independent values or quantities that can vary in an analysis without breaking any constraints. It is a crucial part of calculating the F-value in ANOVA as well as determining the critical value from the F-distribution table. The degrees of freedom for any effect (like package design) are calculated by subtracting one from the number of levels of the factor. For the error, they are computed by reducing the total number of observations by the number of levels in all factors being tested.

The degrees of freedom help shape the F-distribution and affect the critical values needed to make a decision about the statistical significance. In the exercise, calculating the correct degrees of freedom was fundamental to performing the respective analyses on package designs and geographical areas, guiding the manufacturer to conclude that neither factor significantly influenced the sales.