• Basic Statistics Analysis

• Regression Analysis

• Discriminant Analysis

• ANOVA / MANOVA

• Factor Analysis

One:
Do a correlated (paired) t-test comparing the Brand Index Score (summed Likerts or semantic differentials) for Brand A (pick any of the three brands for which ads were tested--this will be referred to as "Brand A" or "B" or "C" throughout these problems) with the Brand Index Score (summed Likerts or sd’s) for Brand B. Do this for all combinations of the three test brands. (Note the test brands are those for which the respondents saw advertisements; brand index score refers to a summation for each respondent over those nine or ten or eleven items which refer to the brand as the measurement object.)

Paired t-tests for Brand Index Score
Table 1:  Mean Brand Index Scores for all brands

Sample Size = 66

Table 2: Paired t-Tests of Brand Index Scores

Sample Size = 66
* p  <  0.15

In an attempt to determine respondents’ feelings towards each of the three test brands, Nabisco, Orville, and Pepperidge Farm, a brand index score was calculated for each. By examining the responses to the 10 Likert items, which appeared on the survey, a brand index score was calculated. The Scores ranged from 10 to 50, indicating less favorable to favorable feelings towards the brands. After comparing the three brand index scores it was apparent that all three resulted in only mildly positive feelings. Each of the three brand means fell into the low 30’s, which is only slightly above the mid-point of 25.

Following this, a paired t-test was conducted in order to determine whether or not the results could be accurately projected to a larger population. The paired t-test shows that of the three brand pairs none would project to a larger population. In 85 or more samples out of every 100 samples, none of the three brand pairs would result in similar mean brand index scores.

Two:
Do two between-groups t-tests where the groups are (1) those respondents who moved up pre-to-post exposure to the advertisements on the constant sum scale on Brand A and (2) those who moved down pre-to-post on Brand A; the dependent variables used should be (1) the Brand Index Score for   Brand A and (2) either an Ad Index Score for the advertisement for Brand A (if your data have these measurements--scalar Likert or sd responses to the ad) or the number of 'favorable' items checked from an ad checklist for Brand A (if you have a checklist format rather than a scalar format for ad response).

Nabisco Mover Statistics

Sample Size Up = 16
Sample Size Down = 15

Between t-Tests for Nabisco

* p  <  0.15

There was no significant difference (p < .15) for both the Brand and Ad Index Scores between those respondents whose opinion of the brand moved up or down on the after viewing the advertisement. 
The Ad Index Score was found for Nabisco by calculating all the positive statements related to the ad and indicated by respondents. The score had a range of 0 to 15. Using the Ad Index Score and the Brand Index Score, between-groups t-tests were conducted. These results were then used to determine if the findings for the Nabisco ad could be projected to a larger population.
Between-groups t-test was conducted in an attempt to determine whether the Nabisco ad shown in the survey may have had an effect on respondents. The post-exposure brand index score was calculated and compared to the pre-exposure brand index score. Up movers were defined by an increase in the brand index score while down movers were characterized by a decrease in the score. In total, 16 of the respondents moved up after being exposed to the Nabisco ad while 15 of those who responded moved down. It can be assumed that 31 of the total 66 respondents changed their opinion (moved up or down) after being exposed to the Nabisco advertisement.

Three:
Do a chi-squared significance test to find out if there is a significant relationship between up, same and down pre/post movers on Brand A and being above or below the median Brand Index Score on Brand A.

Chi Squared Test – Nabisco

Sample Size = 66

Chi Square Value = 2.59*
* p  <  0.15

Note: 0 cells (.0%) have expected count less than 5. the minimum expected count is 5.81

Four above the median respondents moved up after exposure to the Nabisco ad, representing approximately 13.3% of all respondents above the median.33.3% of all up movers, 6.5% of all respondents. Twenty respondents above the median remained the same, representing approximately 66.7% of all respondents above the median, 57.1% of all respondents remaining the same, and 32.3% of all respondents. Finally six respondents above the median moved down, representing approximately 20.0% of all above the median respondents, 40.0% of all respondents moving down, and 9.7% of total respondents.
The Chi-Square test also showed that while analyzing the median respondents, 8 moved up, representing 25.0% of all respondents below the median, 66.7% of all down movers, and 12.9% of all respondents. Fifteen respondents below the median remained the same after being exposed to the ad, representing approximately 46.9% of all respondents above the median, 42.9% of all respondents remaining the same, and 24.2% of all respondents. Lastly, regarding the respondents below the median who moved down, nine respondents below the median moved down, representing approximately 28.1% of all below median respondents, 60.0% of all respondents moving down, and 14.5% of total respondents.
This chart suggests that there was a similar effect on those who liked and those who disliked the advertisement. Four respondents moved up in the above median group while eight moved up in the below median group. The chart also suggests that there was a similar negative effect on the respondents. Six of the respondents moved down in the above median group while nine moved down in the already below median group.
The test illustrates that these results are significant at the 0.15 level and that in 85 or more samples out of every 100 samples drawn from the same population as this sample, the same results for the Nabisco ad could be expected for a larger population.

Four:

For each of the three test brands, i.e., Brands A, B, and C, find out how many people (using the entire sample) go up, down, or stay the same pre-to-post exposure on the constant-sum scale by doing, e.g.:

                     If PreCoke < PostCoke then let move=1 (Up)
                     If PreCoke = PostCoke then let move=2 (Same)
                     If PreCoke > PostCoke then let move=3 (Down)

Pre to Post Test Exposure Movement

Sample Size = 66
 
While the majority of the respondents stayed the same after being exposed to the advertisements, a little less than half moved up or down. A positive change (up move) to the advertisements was similar between Nabisco, with 16 people and Pepperidge Farm, with 12 people, while Orville had the most number of positive responses among 22 respondents. A negative change (down move) to the advertisements was identical between Orville and Pepperidge Farm, with 12 people, while Nabisco had the most number of negative responses among 15 respondents.
These results indicate that among the three test brands, Orville had the most number of respondents to change their opinion in a positive direction after seeing the advertisements. The results also show that the Pepperidge Farm brand had the exact number of people to respond positively as to negatively after seeing the ad. 

Five:

For Brand A, add ten Likerts (or sd's if you have them) toward the brand into an index (we have been calling this the Brand Index Score). For Brand B, add ten Likerts (or however many you might have) for this brand into an index. How many of the total number of respondents you have in your data have a higher Brand A Index Score than
Brand B Index Score?

For example:
Let BA = v10+vl1+v12 and so on
Let BB = v40+v41+v42...
If BA > BB then let pass=1
If BA <= BB then let pass=2
Then tabulate (frequency count on the new variable "pass" to find the answer.

Count of higher Brand Index Scores for Nabisco – Orville

Sample Size = 66
In the above table, after subtracting the Brand Index Scores of Orville from the Brand Index Scores of Nabisco, 19 of the 66 respondents had a higher Brand Index Score for Nabisco than they did for Orville.
28.8% of the respondents Nabisco higher than Orville after viewing the ad.  This means that 19 out of 66 respondents reported higher scores on the summed Likert responses for Nabisco than they did for Orville. 

Six:
Calculate the simple correlation coefficient between Brand A Index Scores and Brand B Index Scores.

Correlations

Sample Size = 66
* p  <  0.15

According to the negative correlation, illustrated in the 0.69 correlation coefficient, this correlation suggests that those respondents who like the Nabisco Brand tend to not like the Orville brand. 

Seven:

Do any type of basic statistical analysis you want involving one of the following which does something different than any of the above problems required you to do: frequency counting, chi-square, t-test (between-groups or paired), standard scores, simple correlation. The only requirement for this problem is that you must "select" a subset of the respondents in your data set on which to conduct any analyses, i.e., do not conduct the analyses on all subjects in your data file.

This chart shows the simple correlation coefficient between those respondents who often purchased health or diet food and those who exercise frequently.

Correlations

Sample Size = 66
* p  <  0.15

The 1.0 correlation coefficient suggests a positive relationship between those respondents who often purchased health or diet food and those who exercise frequently.

Regression Analysis (Simple and Multivariate)

For the following 3 cases a multiple linear regression analysis was performed for each of the three brands of 100 calorie packaged snack foods, Nabisco, Orville, and Pepperidge Farm. The analysis was intended to look at the respondent’s feelings towards the brand, and their opinion of the ad and determine if there is a relationship. To determine the respondent’s feelings towards a particular brand the ten Likert items were taken into consideration and used as the independent variables for the linear regression. The dependent variable was the brand change score, which was calculated by taking the pre-exposure opinion and subtracting it from the post-exposure opinion. 
As shown in the Linear Regression results, the R column represents the relationship between the respondent’s feelings for the brand and their opinion of the brand’s ad. A strong positive relationship would be represented by a 1.0, while a moderate relationship would be represented by a 0.5. In the following analysis the Nabisco brand relationship resulted in a 0.5, suggesting that the respondent’s feelings for the brand Nabisco, and their opinion of the brand’s ad have a moderate relationship. For the second brand, Orville, the R value of 0.6 would suggest that a slightly higher than moderate relationship is the result. The third brand, Pepperidge Farm resulted in the lowest correlation, with an R value of 0.3.

These Brand Index Items resulted in a moderate, positive correlation with the Change Score.  This correlation means that only 20.5% of how much the respondents liked the ad is explained by how much they liked the brand.  The F value is 1.4 and is therefore not significant, meaning that the correlations apply to this sample only, and cannot be generalized to other populations.

These Brand Index Items for the second brand, Orville, resulted in higher, yet still relatively moderate, positive correlation with the Change Score. This correlation showed that only 30.6% of how much the respondents liked the ad is explained by how much they liked the brand.  The F value is 2.4 and is therefore not significant, meaning that the correlations apply to this sample only, and cannot be generalized to other populations.

The Brand Index Items for Pepperidge Farm yielded the lowest positive correlation with the Change Score out of the three brands. Only 9.8 of how much the respondents liked the ad was defined by how much they liked the brand.  The F value is 0.60 and is therefore not significant, meaning that the correlations apply to this sample only, and cannot be generalized to other populations.

In the analysis above some of the variables were positively associated with the change score and will move up as the change score does. These include: Taste Good, Trust Brand, and High Quality. Other variables were negatively associated with the change score and will move up as the score moves down. These include: Good Snack Food, Good Value, Too Expensive, Would not recommend, Loyal to Brand, and Would Not Buy. In the sample above only Trust Brand is significant at the p < .15 level, meaning that in 85 or more samples drawn from the same population as this sample that only this correlation would be about what it is in this sample for that item.

For the second brand, Orville, Taste Good, Good Value, Not Recommend, and High Quality were positively associated with the change score and will move up as the change score does. Good Snack Food, Too Expensive, Trust Brand, Healthy Choice, Loyal to Brand, and Would Not Buy is negatively associated with the change score and will move up as the score moves down. In this sample, Would Not Recommend is significant at the p < .15 level, meaning that in 85 or more samples drawn from the same population as this sample that only this correlation would be about what it is in this sample for that item.

Finally, in the analysis shown for the third brand, Pepperidge Farm, Taste Good, Good Value, Not Recommend, and High Quality were positively associated with the change score and will move up as the change score does. Good Snack Food, Too Expensive, Trust Brand, Healthy Choice, Loyal to Brand, and Would Not Buy is negatively associated with the change score and will move up as the score moves down. In this sample, High Quality is significant at the p < .15 level, meaning that in 85 or more samples drawn from the same population as this sample that only this correlation would be about what it is in this sample for that item.

Discriminant Analysis
The following results are those of a discriminant analysis on the Pepperidge farm brand. Subjects are split into "up" and "down" to make two groups of subjects for the categorical dependent variable. The independent variables are the ten Likert items for that brand.

A discriminant analysis was conducted for the Pepperidge Farm Brand in an attempt to determine which brand characteristics, if any, are most influential between those respondents who moved up and down, in regards to their opinion of the brand, after seeing the ad.  The up movers are those respondents who showed increased brand index scores after seeing the Pepperidge Farm ad. Likewise the Down movers were those who had a lower brand index score. 12 respondents were up movers and 12 were down movers, out of a total of 66 respondents. The ten brand Likert items were used as the independent variable while the movers were used as a dependent variable.
The results above indicate a mean range of 1.8 to 4.4 for those who moved up and a mean range of 1.6 to 4.5 for those who moved down, indicating similar ranges in both groups. Also in both groups the Likert item, would not buy was significantly lower that the rest of the items, suggesting that this particular item did not have a significant impact on respondents.

The significant difference between the up movers and down movers in the group centroid scores is 1.92 signifying that the effect of the brand characteristics, represented by the Likert scores, have an effect discriminating between the two groups.

The Pepperidge farm Wilks’ Lambda is 0.50, with a Chi-squared of 11.85 and 10 degrees of freedom. These results point to a significance in the Wilks’ Lambda score, meaning that 85 out of every 100 samples drawn from the same sample population as these 66 respondents, similar results can be expected.

The highest influence on the post ad exposure brand index scores is defined by the standardized discriminant coefficients. These were used specifically in this case to determine the important explanatory variables for the brand Pepperidge Farm.

The discriminant coefficients are used to show how well an increase or decrease in brand index scores following ad exposure may be predicted. In the case of Pepperidge farm only two of the Likert items, taste good, and high quality, can be used to predict a shift in the Pepperidge farm index score.
Discriminant Equation for Pepperidge Farm:

Move score for Pepperidge Farm = -11.5 -3.8(Good Snack Food) + 4.2(Taste Good) – 2.0(Good Value) –0.0(Too Expensive) – 1.4(Trust Brand) + 0.9(Would Not Recommend) + 4.3(High Quality) +0.1(Healthy Choice) +0.0(Loyal to Brand) + 1.1(Would not Buy)

t-ratio = (% correctly classified – 0.05)/ √ % correctly classified (1- % correctly classified)/ Total Sample + (0.5)(0.5)/Total Sample
            = (0.75- 0.5)/ √(0.75)(0.25)/ 24 + (0.5)(0.5)/ 24
            = 1.85

The classification results indicate that 75% of grouped cases were correctly classified. 9 of the total 12 up movers and 9 of the total 12 down movers were classified correctly. The remaining 3 in each group were incorrectly classified. The t-ratio is calculated to be 1.85.

ANOVA / MANOVA

A two-way analysis of variance, or ANOVA test, was conducted using the move score (the respondents change in opinion after being exposed to the ad) as the independent variable, while the number of times the respondent’s exercised per week was used as the dependent variable. The number of respondents for both variables was 66.

The number of up and down movers was similar in the chart above with mean scores of 3.7 and 3.6 for the 0-2 exercisers and 4.2 and 4.6 for the 3+ exercisers. Also the difference in exercise for the up movers was only 0.5 while the difference in exercise for the down movers was 1.0.

  In the ANOVA table the F-ratio is used to test significance. In this particular test the score for the between main effect of exercise (0.01) was found to be significant at the 0.15 level, meaning that in 85 out of every 100 samples drawn from the same sample population as the 66 respondents we can expect similar results, and can then project them to a larger population.

A multivariate analysis of variance, or MANOVA, was conducted on the Pepperidge Farm brand. The independent variables were the move score and the number of times per month the respondents exercised, while the dependent variables were the 10 Likert items. The number of respondents for both variables was 66.
         

The result of the test were similar mean scores in both up and down movers, as well as the scores between those who exercise 0 to 2 times a week and those who exercise 3 or more times. The mean score for the up and down movers who exercise 3+ times per week were identical (4.6), suggesting that those whose opinion that the brand is a good snack moved down and up regardless of exercise habits. The mean score for down movers (4.1) for trust brand, high quality and not recommend the brand was significantly lower suggesting that those down movers who exercised 0 to 2 times per week still did not like or trust the brand after seeing the ad.
The MANOVA table above illustrates that none of the f-ratios are significant and could therefore not be projected to a larger population.

Factor Analysis

A factor analysis was conducted for all three brands of 100-calorie packaged snack foods. In doing this the related brand attributes, or the 10 Likert items, were placed in factor groups. Following this, a brand attitude score was calculated, and then used to conduct a paired t-test in order to determine their significance.

For the brands Nabisco and Orville Redenbacher, only one of the factors, number one had an Eigenvalue > 1, which would explain the variance of at least one of the Likert items. The factor for Nabisco accounts for 80.9% of the variance, leaving the remaining 11% unexplained. For Orville Redenbacher, the factor accounts for 76.6% of the variance, leaving the remaining 24.4% unexplained. In the case of Pepperidge Farm, two of the factors, 1 and 2, had an Eigenvalue > 1. Together they account for 80.9% of the variance, leaving the remaining 10.1% unexplained.

The Factor Matrix is used in the remainder of the analysis to develop Nabisco, Orville Redenbacher, and Pepperidge farm’s brand attitude scores.

  In an attempt to determine respondents’ attitudes towards each of the three test brands, Nabisco, Orville, and Pepperidge Farm, an attitude score was calculated for each, by examining the responses to the 10 Likert items, which appeared on the survey. The Scores ranged from 10 to 50, indicating less favorable to favorable attitudes towards each of the three brands. After comparing the three brand index scores it was apparent that all three resulted in only mildly positive attitudes. When looking each of the three brands, the mean scores range from 27 to 30.8 indicating a slight rise above the mid-point of 25.

Following this, a paired t-test was conducted in order to determine whether or not the results could be accurately projected to a larger population. The paired t-test shows that of the three brand pairs, none would project to a larger population. In 85 or more samples out of every 100 samples, none of the three brand pairs would result in similar mean brand attitude scores.