Question

The following experiment was performed to determine the effect of two advertising campaigns on three kinds of cake mixes. Sales of each mix were recorded after the first advertising compaign and then after the second advertising campaign. This experiment was repeated 3 times for each advertising campaign with the following results: $$ \begin{array}{|l|l|l|} \hline & \text { Campaign 1 } & \text { Campaign 2 } \\ \hline \text { Mix 1 } & 574,564,550 & 1092,1086,1065 \\ \hline \text { Mix 2 } & 524,573,551 & 1028,1073,998 \\ \hline \text { Mix 3 } & 576,540,592 & 1066,1045,1055 \\ \hline \end{array} $$ Set up an ANOVA table for this problem and find the appropriate sums of squares, degrees of freedom and mean squares.

Short answer

From the given sales data, we computed the following means and grand mean values: - Mix 1, Campaign 1: 562.67 - Mix 2, Campaign 1: 549.33 - Mix 3, Campaign 1: 569.33 - Mix 1, Campaign 2: 1081 - Mix 2, Campaign 2: 1033 - Mix 3, Campaign 2: 1055.33 - Grand Mean: 808.45 For the ANOVA table, we calculated the sum of squares (SS), degrees of freedom (df), and mean squares (MS) for treatments and error. ANOVA table: $$ \begin{array}{|l|l|l|l|l|} \hline \text{Source of Variation} & \text{Sum of Squares (SS)} & \text{Degrees of Freedom (df)} & \text{Mean Squares (MS)} \\ \hline \text{Treatments} & SSTr & 2 & MSTr \\ \hline \text{Error} & SSE & 15 & MSE \\ \hline \text{Total} & SST & 17 & \\ \hline \end{array} $$

Step-by-step solution

Calculate the means and the grand mean

First, we need to calculate the mean sales for each mix within each campaign, as well as the grand mean (the overall mean across all data points). To do this, add up the sales values for each mix in each campaign, and divide by the number of experiments (3). Then, calculate the grand mean by adding up all mean sales values and dividing by the total number of means (6). Using the given sales data, Mean sales for each mix in Campaign 1: - Mix 1: \((574 + 564 + 550) / 3 = 562.67\) - Mix 2: \((524 + 573 + 551) / 3 = 549.33\) - Mix 3: \((576 + 540 + 592) / 3 = 569.33\) Mean sales for each mix in Campaign 2: - Mix 1: \((1092 + 1086 + 1065) / 3 = 1081\) - Mix 2: \((1028 + 1073 + 998) / 3 = 1033\) - Mix 3: \((1066 + 1045 + 1055) / 3 = 1055.33\) Grand Mean: \((562.67 + 549.33 + 569.33 + 1081 + 1033 + 1055.33) / 6 = 808.45\)

Calculate the total sum of squares (SST)

To find the SST, we need to find the difference between each sales value and the grand mean, square it, and then sum up all these squared differences. SST = \((549 - 808.45)^2 + ...\), continue this process for all nine sales values in Campaign 1 and all nine sales values in Campaign 2.

Calculate the sum of squares due to treatments (SSTr)

SSTr measures the variability in the data due to the different treatments (in this case, the advertising campaigns). To find SSTr, calculate the difference between each mean sales value and the grand mean, square it, and then multiply by the number of experiments (3). Finally, sum up all these squared differences. SSTr = \(3((562.67 - 808.45)^2 + (549.33 - 808.45)^2 + (569.33 - 808.45)^2 + (1081 - 808.45)^2 + (1033 - 808.45)^2 + (1055.33 - 808.45)^2)\)

Calculate the sum of squares due to error (SSE)

SSE measures the variability in the data that cannot be explained by the treatments. To find SSE, subtract SSTr from SST: SSE = SST - SSTr

Calculate the degrees of freedom

Next, we need to find the degrees of freedom (df) for SST, SSTr, and SSE: - df for total sum of squares (SST): \((6 - 1)\text{treatments} \times (3 - 1)\text{block} - 1 = 17\) - df for sum of squares due to treatments (SSTr): \((3 - 1)(2 - 1) = 2\) - df for sum of squares due to error (SSE): \(17 - 2 = 15\)

Calculate the mean squares

Now we need to find the mean square values for SSTr and SSE. To do this, divide each sum of squares by its respective degrees of freedom: - Mean square due to treatments (MSTr): \(SSTr/2\) - Mean square due to error (MSE): \(SSE/15\)

Set up the ANOVA table

Finally, we will set up the ANOVA table with the calculated values: $$ \begin{array}{|l|l|l|l|l|} \hline \text{Source of Variation} & \text{Sum of Squares (SS)} & \text{Degrees of Freedom (df)} & \text{Mean Squares (MS)} \\ \hline \text{Treatments} & SSTr & 2 & MSTr \\ \hline \text{Error} & SSE & 15 & MSE \\ \hline \text{Total} & SST & 17 & \\ \hline \end{array} $$ This table shows the sum of squares, degrees of freedom, and mean squares for each source of variation in the sales data.

Key concepts

Sum of Squares

In the context of ANOVA (Analysis of Variance), the Sum of Squares is a critical value that represents the total variability within the dataset. Its calculation involves summing up all the squared differences between individual observations and the grand mean.

For example, if we want to determine how different advertising campaigns affect cake mix sales, we calculate the sum of squares to quantify the variation in sales figures. We take each sale figure, subtract the overall average (grand mean) of all campaigns, square the result to eliminate negative values, and sum them up for the total sum of squares (SST). This provides insight into the total variation present across all observations.

Degrees of Freedom

Degrees of freedom (df) represent the number of independent values in the data that are free to vary when we are estimating statistical parameters. In simple terms, it’s the count of values that can change without affecting the sample mean.

In the presented advertising campaign problem, degrees of freedom help us to define the independent pieces of information we have from the data. For calculating df, we take into account the number of treatments, blocks, and subtract constraints such as the requirement to calculate the overall mean. They play a crucial role in the subsequent calculations within ANOVA, especially when determining the mean squares and the F-statistic.

Mean Squares

The Mean Squares in ANOVA translate the variability due to specific factors into an average per degree of freedom. This is achieved by dividing the sum of squares by the corresponding degrees of freedom.

For instance, Mean Square due to Treatments (MSTr) is calculated by dividing the Sum of Squares due to Treatments (SSTr) by its degrees of freedom. Similarly, Mean Square due to Error (MSE) is found by dividing the Sum of Squares due to Error (SSE) by its degrees of freedom. These mean squares help in comparing how much variability each source contributes when compared to the variability within the error term.

Treatment Variability

Treatment variability is the portion of the total variance in the data that can be attributed to the differences between the levels of the independent variable, in our case, the two advertising campaigns.

It quantifies the effect of the independent variable on the dependent variable by comparing the means across different treatments. ANOVA helps to assess whether the observed differences in means are significant or could have occurred by chance. It answers the question: 'Do different advertising campaigns result in significantly different sales?'

Error Variability

Error variability, on the other hand, represents the portion of total variability that cannot be explained by treatment effects. It's the natural variability in the data due to random fluctuations, measurement errors, or other sources of variation not accounted for by the independent variable.

In the analysis of our cake mix sales data, error variability is assessed to understand how much of the total variance is due to inherent unpredictability within the experiment. Understanding error variability is essential because it serves as a benchmark for analyzing treatment effects—in essence, it provides the 'noise level' against which the 'signal' of treatment effects is measured.