Questions: The makers of Country Boy Corn flakes are thinking about changing the packaging of the cereal with the hope of improving sales. In an experiment, four stores of similar size in the same region sold the cereal in different shaped boxes to determine if the mean sales are the same, no matter which shape box is used? Cube Cylinder Pyramid Rectangle 117 106 76 168 87 118 68 95 61 178 112 126 93 91 64 81 77 80 89 112 Classify the problem as being a Chi-square test of independence or homogeneity, Chi-square goodness-of-fit, Chi-square for testing or estimating σ^2, F test for two variances, One-way ANOVA, or Two-way ANOVA, then perform the following. - F test for two variances - Chi-square test of independence - One-way ANOVA - Chi-square for testing or estimating σ^2 - Chi-square test of homogeneity. - Two-way ANOVA - Chi-square goodness-of-fit (a) Give the value of the level of significance. 0.05

Determine the results of the One-way ANOVA test for the sales data of different cereal box shapes.

Calculate \(SS_{between}\).

The sum of squares between groups is calculated as follows: \[ SS_{between} = \sum_{i=1}^k n_i (\bar{X}_i - \bar{X})^2 = 4911.75 \]

Calculate \(SS_{within}\).

The sum of squares within groups is calculated as follows: \[ SS_{within} = \sum_{i=1}^k \sum_{j=1}^{n_i} (X_{ij} - \bar{X}_i)^2 = 13561.2 \]

Calculate \(MS_{between}\).

The mean square between groups is calculated as: \[ MS_{between} = \frac{SS_{between}}{df_{between}} = \frac{4911.75}{3} = 1637.25 \]

Calculate \(MS_{within}\).

The mean square within groups is calculated as: \[ MS_{within} = \frac{SS_{within}}{df_{within}} = \frac{13561.2}{16} = 847.575 \]

Calculate the F-statistic.

The F-statistic is calculated as: \[ F = \frac{MS_{between}}{MS_{within}} = \frac{1637.25}{847.575} = 1.9317 \]

Calculate the p-value.

The p-value is calculated as: \[ P = 1 - F(F_{observed}; df_{between}, df_{within}) = 1 - F(1.9317; 3, 16) = 0.1652 \]

The results of the One-way ANOVA test are:

• Degrees of Freedom Between Groups: \(3\)
• Degrees of Freedom Within Groups: \(16\)
• F-Statistic: \(1.9317\)
• P-Value: \(0.1652\)
• Mean Square Between Groups: \(1637.25\)
• Mean Square Within Groups: \(847.575\)

The results indicate that there is no statistically significant difference in mean sales among the different shapes of cereal boxes, as the p-value \(0.1652\) is greater than the significance level \(0.05\).

\(\boxed{\text{No significant difference in mean sales among box shapes.}}\)