91Ó°ÊÓ

A company plans to promote a new product by using one of three advertising campaigns. To investigate the extent of product recognition resulting from the campaigns, 15 market areas were selected, and 5 were randomly assigned to each campaign. At the end of the campaigns, random samples of 400 adults were selected in each area, and the proportions who indicated familiarity with the product appear in the following table. $$\begin{array}{ccc}\hline & {\text { Campaign }} \\\\\hline 1 & 2 & 3 \\\\\hline .33 & .28 & .21 \\\\.29 & .41 & .30 \\\\.21 & .34 & .26 \\\\.32 & .39 & .33 \\\\.25 & .27 & .31 \\\\\hline\end{array}$$ a. What type of experimental design was used? b. Is there sufficient evidence to indicate a difference in locations of the distributions of product recognition scores for the three campaigns? Bound or give the approximate \(p\) -value. c. Campaigns 2 and 3 were, respectively, the most and least expensive. Is there sufficient evidence to indicate that campaign 2 is more successful than campaign 3? Test using the Mann-Whitney \(U\) procedure. Give the associated \(p\) -value.

Step by step solution

01

Identify the Experimental Design

The problem describes an experiment where 15 market areas are selected, with 5 market areas randomly assigned to each of the three advertising campaigns. This scenario aligns with a Completely Randomized Design because each experimental unit (in this case, each market area) is randomly assigned to one of the treatment groups (campaigns) without any prior consideration of their characteristics.

02

Set Up Hypothesis for Kruskal-Wallis Test

To determine if there is statistical evidence of a difference in product recognition across the three campaigns, we perform the Kruskal-Wallis test. This non-parametric test assesses whether there is a difference in the distributions of product recognition scores among the three campaigns.- Null Hypothesis, \(H_0\): There is no difference in the distributions of product recognition scores for the three campaigns.- Alternative Hypothesis, \(H_1\): At least one campaign has a different distribution of product recognition scores.

03

Calculate Ranks for Campaign Scores

Assign ranks to all the recognition proportions, treating them as a single dataset. If there are ties, assign the average rank to the tied scores.

04

Compute Kruskal-Wallis Test Statistic

Using the ranks computed in the previous step, compute the Kruskal-Wallis test statistic \( H \). This involves using the formula:\[H = \frac{12}{N(N+1)} \sum_{i=1}^k \frac{R_i^2}{n_i} - 3(N+1)\]where \( k \) is the number of groups (campaigns), \( N \) is the total number of observations, \( R_i \) is the sum of the ranks for group \( i \), and \( n_i \) is the number of observations in group \( i \).

05

Determine Approximate p-value for Kruskal-Wallis Test

Compare the computed \( H \) statistic to the Chi-square distribution with \( k-1 \) degrees of freedom (where \( k \) is the number of groups). The p-value can be obtained using statistical tables or software.

06

Set Up Hypothesis for Mann-Whitney U Test

To compare just the scores between campaigns 2 and 3, use the Mann-Whitney U test. - Null Hypothesis, \(H_0\): There is no difference in the scores between campaigns 2 and 3.- Alternative Hypothesis, \(H_1\): Campaign 2 has higher scores than campaign 3.

07

Compute Mann-Whitney U Statistic

Rank the combined scores from campaigns 2 and 3 together. Calculate the sum of the ranks for each campaign. Use these to calculate the U statistic with the formula:\[U = n_1n_2 + \frac{n_1(n_1+1)}{2} - R_1\]where \( n_1 \) is the number of observations in campaign 2, \( R_1 \) is the rank sum for campaign 2, and \( n_2 \) is the number of observations for campaign 3.

08

Determine p-value for Mann-Whitney U Test

Using the U value calculated, determine the p-value from the Mann-Whitney U distribution. This can be done using statistical tables or computational software.

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Key Concepts

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

Completely Randomized Design

In experimental studies, a completely randomized design is a fundamental approach where subjects are randomly allocated to different treatment groups. This randomization helps ensure that differences in outcomes can be attributed to the treatments themselves, rather than other extraneous factors. In our case study, the experiment involved selecting 15 market areas, with 5 areas randomly assigned to each of the three advertising campaigns.
This type of design is ideal when you want to minimize biases and make fair comparisons between groups. It's particularly useful when the experimental units, the market areas in this scenario, do not have any distinctions influencing the variable of interest (here, product recognition).

• Each group or treatment is completely independent from the others.
• Random assignment of units helps control for confounding variables.

This simplicity is a key feature of the completely randomized design, making interpretation straightforward and powerful, especially when comparing results across different treatment conditions.

Kruskal-Wallis Test

The Kruskal-Wallis test is a non-parametric statistical test used when you want to compare three or more independent groups to see if there are differences in their distributions. It's especially handy when the data does not follow a normal distribution or when dealing with ordinal-level data. In the context of our exercise, it helps determine if there's a significant difference in product recognition scores across the three advertising campaigns.
Here's how the test works:

• Rank all data from all groups together.
• Sum the ranks for each group, which forms the basis for the test statistic.
• Calculate the Kruskal-Wallis test statistic, H, using the ranks.

This method essentially tests the null hypothesis that the groups come from the same distribution against the alternative hypothesis that at least one group differs. The test statistic is then compared to a Chi-square distribution to find the p-value, which helps in accepting or rejecting the null hypothesis.
The Kruskal-Wallis test is robust and less sensitive to outliers, making it a versatile tool in experiments involving non-normal data.

Mann-Whitney U Test

The Mann-Whitney U test is another non-parametric tool, but it focuses on comparing differences between just two groups. This makes it similar to a t-test, but without the requirement of normally distributed data. In our scenario, it is used to compare the success of advertising campaigns 2 and 3 directly, which are noted as the most and least expensive, respectively.

• The null hypothesis states that there is no difference between the two campaigns’ scores.
• The alternative hypothesis suggests that campaign 2 has a higher score than campaign 3.

Every observation from both campaigns is ranked, and then the ranks are summed for each group. The Mann-Whitney U statistic is calculated using these ranks. From there, one determines the p-value, which tells us if the observed difference is statistically significant.
This test is particularly useful in studies where the data has non-normal distributions or when the sample size is small. Its reliance on ranked rather than raw data makes it resilient to skewness and outliers.