Question

How satisfied are you with your current mobile-phone service provider? Surveys done by Consumer Reports indicate that there is a high level of dissatisfaction among consumers, resulting in high customer turnover rates. \({ }^{10}\) The following table shows the overall satisfaction scores, based on a maximum score of \(100,\) for four wireless providers in four different cities. $$ \begin{array}{lcccc} & & & & \text { San } \\ & \text { Chicago } & \text { Dallas } & \text { Philadelphia } & \text { Francisco } \\ \hline \text { AT\&T Wireless } & 63 & 66 & 61 & 64 \\ \text { Cingular Wireless } & 67 & 67 & 64 & 60 \\ \text { Sprint } & 60 & 68 & 60 & 61 \\ \text { Verizon Wireless } & 71 & 75 & 73 & 73 \end{array} $$ a. What type of experimental design was used in this article? If the design used is a randomized block design, what are the blocks and what are the treatments? b. Conduct an analysis of variance for the data. c. Are there significant differences in the average satisfaction scores for the four wireless providers considered here? d. Are there significant differences in the average satisfaction scores for the four cities?

Short answer

Answer: To determine if there are significant differences in the average satisfaction scores for the four wireless providers and the four cities, we would compare the F-ratio to the critical F-value for both treatments and blocks. If the calculated F-ratios are greater than their respective critical F-values, then we can conclude that there are significant differences in the average satisfaction scores both for the wireless providers and the cities.

Step-by-step solution

(Step 1: Identifying the Experimental Design)

In this case, the experimental design is a randomized block design, where the blocks are the cities (Chicago, Dallas, Philadelphia, San Francisco) and the treatments are the wireless providers (AT&T, Cingular, Sprint, Verizon).

(Step 2: Organizing the Data for ANOVA)

Before calculating the ANOVA, we need to find the total satisfaction scores for each block (city), treatment (provider), and the grand total. We also need to find the square of the sum of each block (city), treatment (provider), and the grand total. Additionally, we need to calculate the sum of squares of each data point.

(Step 3: Calculating the ANOVA Table Values)

To perform the ANOVA calculations, we need to find the following values for source of variation, sum of squares (SS), degrees of freedom (df), mean square (MS), and F-ratio: - Source of Variation: - Between blocks - Between treatments - Within treatments - Total variation - SS for each source of variation: - SS between blocks - SS between treatments - SS within treatments - SS total - df for each source of variation: - df between blocks = number of blocks - 1 - df between treatments = number of treatments - 1 - df within treatments = (total number of observations) - (number of blocks) - (number of treatments) + 1 - df total = total number of observations - 1 - MS for each source of variation (except total): - MS between blocks = SS between blocks / df between blocks - MS between treatments = SS between treatments / df between treatments - MS within treatments = SS within treatments / df within treatments - F-ratio for each source of variation (except total): - F-ratio between blocks = MS between blocks / MS within treatments - F-ratio between treatments = MS between treatments / MS within treatments

(Step 4: Interpreting the Results)

(a) As identified in step 1, the experimental design used is a randomized block design, with the blocks being the cities and the treatments being the wireless providers. (b) Based on step 3, we can calculate the ANOVA table values for sum of squares, degrees of freedom, mean square, and F-ratio. (c) To determine if there are significant differences in the average satisfaction scores for the four wireless providers, we need to compare the F-ratio between treatments to the critical F-value from the F-distribution table using the degrees of freedom between treatments and within treatments at a given significance level (such as 0.05). If the calculated F-ratio is greater than the critical value, we can conclude that there are significant differences between the satisfaction scores of wireless providers. (d) To determine if there are significant differences in the average satisfaction scores for the four cities, we need to compare the F-ratio between blocks to the critical F-value from the F-distribution table using the degrees of freedom between blocks and within treatments at a given significance level (such as 0.05). If the calculated F-ratio is greater than the critical value, it would indicate that there are significant differences in the satisfaction scores between the cities.

Key concepts

Analysis of Variance

Analysis of Variance (ANOVA) is a statistical method that is 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 is commonly used in experiments where the variables can be split into different categories or groups.

In the context of the survey on mobile-phone service provider satisfaction, ANOVA can help us determine if there are noticeable differences in customer satisfaction scores amongst different wireless providers or across cities. By quantifying the variability within group means and between group means, ANOVA provides a systematic way to assess whether observed differences are due to random chance or true differences in customer satisfaction.

Experimental Design

Experimental design refers to the structure of an investigation where the researcher manipulates one variable to determine if it causes an effect on another variable, while controlling other variables as much as possible. In the exercise, the randomized block design is employed, which is a type of experimental design used to control for known sources of variability between subjects - in this case, the cities.

The 'blocks' in this context are the cities (Chicago, Dallas, Philadelphia, San Francisco). These blocks help to control for variation in customer satisfaction that might occur because of geographic or demographic differences. The 'treatments' are the wireless providers (AT&T, Cingular, Sprint, Verizon) whose effects on customer satisfaction are being compared. By using this design, the researchers can focus on the specific differences in satisfaction levels due to the providers, while controlling for the variability between different cities.

ANOVA Calculations

ANOVA calculations involve several components which need to be computed to draw conclusions from the data. These include Sum of Squares (SS) for each source of variation, degrees of freedom (df), Mean Squares (MS), and the F-ratio.

SS is a measure of total variation within a dataset and is partitioned into components related to the effects of different sources (blocks, treatments, and error). For df, it's a way of quantifying the number of independent pieces of information in the data that can be used to estimate variance. MS represents the average variation within each source and is calculated by dividing SS by its respective df. Lastly, the F-ratio is used to test the significance of the observed ratios between variances, derived by dividing MS of the treatment effects by the MS of the error.

By interpreting these values through the F-distribution, which considers both between-treatments and within-treatments df, researchers can determine if there are significant differences in satisfaction scores. For example, if the F-ratio for service providers is high and exceeds the critical value from the F-distribution table at a given significance level (like 0.05), it suggests that satisfaction varies significantly between providers.

Customer Satisfaction Scores

Customer satisfaction scores are numerical assessments that quantify a customer's level of satisfaction with a service or product. These scores can range from very dissatisfied to very satisfied, and they provide a simple way for businesses to gauge customer sentiment and overall service performance.

In the exercise, satisfaction scores were collected for different wireless providers across various cities. Such scores are crucial because they can influence customer retention and brand reputation. In the ANOVA analysis, these scores are treated as response variables, and the goal is to understand if different providers are delivering varying levels of satisfaction and if this varies by location. The findings can provide actionable insights for companies to enhance their services or tailor them to specific market needs.