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Will improving customer service result in higher stock prices for the companies providing the better service?"When a company's satisfaction score has improved over the prior year's results and is above the national average (currently 75.7 ), studies show its shares have a good chance of outperforming the broad stock market in the long run" (Business Week, March 2, 2009 . The following satisfaction scores of three companies for the 4 th quarters of 2007 and 2008 were obtained from the American Customer Satisfaction Index. Assume that the scores are based on a poll of 60 customers from each company. Because the polling has been done for several years, the standard deviation can be assumed to equal 6 points in each case. \(\begin{array}{lcc}\text { Company } & \text { 2007 Score } & \text { 2008 Score } \\ \text { Rite Aid } & 73 & 76 \\\ \text { Expedia } & 75 & 77 \\ \text { J.C. Penney } & 77 & 78\end{array}\) a. For Rite Aid, is the increase in the satisfaction score from 2007 to 2008 statistically significant? Use \(\alpha=.05 .\) What can you conclude? b. Can you conclude that the 2008 score for Rite Aid is above the national average of \(75.7 ?\) Use \(\alpha=.05\) c. For Expedia, is the increase from 2007 to 2008 statistically significant? Use \(\alpha=.05\) d. When conducting a hypothesis test with the values given for the standard deviation, sample size, and \(\alpha,\) how large must the increase from 2007 to 2008 be for it to be statistically significant? e. Use the result of part (d) to state whether the increase for J.C. Penney from 2007 to 2008 is statistically significant.

Step by step solution

01

Set Up Hypotheses for Part (a)

We want to test if the satisfaction score for Rite Aid has increased significantly from 2007 to 2008. The null hypothesis ( H_0 ) is: μ_1 = μ_2 (no increase in satisfaction score). The alternative hypothesis ( H_a ) is: μ_1 < μ_2 (the satisfaction score has increased). We will use a significance level of α = 0.05.

02

Calculate the Test Statistic for Part (a)

The test statistic is calculated using the formula:\[z = \frac{\bar{x}_2 - \bar{x}_1}{\sigma/\sqrt{n}}\]where \(\bar{x}_1 = 73\), \(\bar{x}_2 = 76\), \(\sigma = 6\), and \(n = 60\).Plugging in these values gives:\[z = \frac{76 - 73}{6/\sqrt{60}} = 3.162\]

03

Determine the Critical Value for Part (a)

For a one-tailed test with α = 0.05 , the critical z-value is approximately 1.645 . If the test statistic is greater than the critical value, we reject the null hypothesis.

04

Decision for Part (a)

The calculated z-value of 3.162 is greater than the critical value of 1.645 . Thus, we reject the null hypothesis and conclude that the increase in Rite Aid's satisfaction score is statistically significant.

05

Hypotheses for Part (b)

We test if the 2008 score for Rite Aid is above the national average. The null hypothesis ( H_0 ) is: μ_2 = 75.7 . The alternative hypothesis ( H_a ) is: μ_2 > 75.7 (2008 score is above the national average).

06

Calculate the Test Statistic for Part (b)

The test statistic is calculated as:\[z = \frac{76 - 75.7}{6/\sqrt{60}} = 0.387\]

07

Decision for Part (b)

The z-value 0.387 is less than the critical value 1.645 , so we fail to reject the null hypothesis. Hence, we cannot conclude that the 2008 score for Rite Aid is statistically above the national average.

08

Set Up Hypotheses for Part (c)

We want to test if the satisfaction score for Expedia has increased significantly. The null hypothesis ( H_0 ) is: μ_1 = μ_2 . The alternative hypothesis ( H_a ) is: μ_1 < μ_2 .

09

Calculate the Test Statistic for Part (c)

Using the formula:\[z = \frac{77 - 75}{6/\sqrt{60}} = 2.581\]

10

Decision for Part (c)

The calculated z-value of 2.581 is greater than the critical value 1.645 . Therefore, we reject the null hypothesis and conclude that the increase in Expedia's satisfaction score is statistically significant.

11

Determine the Required Increase for Significance in Part (d)

To find how large an increase must be for it to be statistically significant, set z = 1.645 (for α = 0.05) and solve for the increase:\[1.645 = \frac{\bar{x}_2 - \bar{x}_1}{6/\sqrt{60}}\]Solving gives an increase of at least 1.27 points.

12

Evaluate Part (e) for J.C. Penney

The increase for J.C. Penney from 2007 (77) to 2008 (78) is 1 point. As 1 point is less than the required 1.27 points for significance, the increase is not statistically significant.

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Key Concepts

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

Statistical Significance

Statistical significance is a fundamental concept in hypothesis testing used to determine whether the observed differences in data are due to chance or actual effects. In the context of customer satisfaction scores, we use statistical significance to ascertain if changes in scores across years reflect genuine improvements in service or merely random fluctuations.
To evaluate statistical significance, we calculate a test statistic, often a z-score, and compare it with a critical value from a statistical table. If the test statistic exceeds the critical value, the change is considered statistically significant.

• This means there is strong evidence against the null hypothesis, suggesting the changes observed are unlikely to be due to random variability alone.
• However, if the statistic does not exceed the critical value, we fail to reject the null hypothesis, suggesting changes might be due to random variance.

Customer Satisfaction

Customer satisfaction refers to consumers' perception of their experience with a product or service. It's a vital measure as it often correlates with business outcomes, such as customer loyalty, repeat purchases, and ultimately, company profitability.
Satisfied customers are more likely to recommend a company, influencing new customer acquisitions and increasing market share. In the exercise provided, customer satisfaction scores were tracked from 2007 to 2008 to assess changes in service quality.
Some key points include:

• Consistent tracking helps enterprises fine-tune strategies for customer retention.
• Improvements in customer satisfaction scores may signal enhancements in service offerings or brand perception.
• Analyzing these scores over time provides insights into customer behavior and potential areas for operational improvement.

American Customer Satisfaction Index

The American Customer Satisfaction Index (ACSI) provides a standardized measure of customer satisfaction across different sectors and companies in the United States. It is widely used because it provides information on how customers perceive the quality of products and services offered by different companies.
The ACSI employs random sampling to survey customers and collects their feedback about various aspects of their experiences. This data is then used to create a satisfaction score on a scale generally ranging from 0 to 100.

• ACSI scores are valuable in gauging changes in customer satisfaction over time and comparing performance with other companies or industries.
• A higher ACSI score indicates better customer satisfaction, which can lead to competitive advantages in terms of customer loyalty and brand reputation.
• For the companies discussed in the exercise, the ACSI helps determine whether their service changes over a year are meaningful in the eyes of customers.

Null and Alternative Hypotheses

The null and alternative hypotheses form the foundation of hypothesis testing. The null hypothesis ( H_0 ) represents the default assumption that no effect or difference is present. In contrast, the alternative hypothesis ( H_a ) suggests the presence of an effect or difference.
In hypothesis testing, we always test these hypotheses against each other:

• The null hypothesis is assumed to be true until there is sufficient evidence to reject it in favor of the alternative hypothesis.
• For example, in the exercise, the null hypothesis posits that there is no increase in customer satisfaction scores between 2007 and 2008, while the alternative hypothesis proposes that there is an improvement.
• Rejection of the null hypothesis indicates statistical significance, giving us the confidence that observed changes are likely not due to random chance.