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Virtual call centers are staffed by individuals working out of their homes. Most home agents earn \(\$ 10\) to \(\$ 15\) per hour without benefits versus \(\$ 7\) to \(\$ 9\) per hour with benefits at a traditional call center (BusinessWeek, January 23, 2006). Regional Airways is considering employing home agents, but only if a level of customer satisfaction greater than \(80 \%\) can be maintained. A test was conducted with home service agents. In a sample of 300 customers 252 reported that they were satisfied with service. a. Develop hypotheses for a test to determine whether the sample data support the conclusion that customer service with home agents meets the Regional Airways criterion. b. What is your point estimate of the percentage of satisfied customers? c. What is the \(p\) -value provided by the sample data? d. What is your hypothesis testing conclusion? Use \(\alpha=.05\) as the level of significance.

Step by step solution

01

Formulate Hypotheses

To determine if customer service from home agents satisfies Regional Airways’ requirement, we set up hypotheses. The null hypothesis (\( H_0 \)) is that customer satisfaction is 80% or less. The alternative hypothesis (\( H_a \)) is that satisfaction is greater than 80%. \[ H_0: p \leq 0.80 \quad \text{(customer satisfaction is 80% or less)} \] \[ H_a: p > 0.80 \quad \text{(customer satisfaction exceeds 80%)} \]

02

Calculate Point Estimate

The point estimate for the proportion of satisfied customers is calculated as the ratio of the number of satisfied customers to the total number surveyed.\[ \hat{p} = \frac{252}{300} = 0.84 \]So, the point estimate is 0.84 or 84%.

03

Find the Test Statistic

To test the hypothesis, a z-test for proportions is used. The formula for the test statistic is: \[ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}} \] Substitute \( \hat{p} = 0.84\), \( p_0 = 0.80\), and \( n = 300 \): \[ z = \frac{0.84 - 0.80}{\sqrt{\frac{0.80 \times 0.20}{300}}} \approx \frac{0.04}{0.0231} \approx 1.73 \]

04

Determine p-value

Using the calculated z-value of 1.73 and looking up the corresponding p-value in a standard normal distribution table, we find: The p-value corresponding to \( z = 1.73 \) is approximately 0.0418.

05

Make a Decision

We compare the p-value to the significance level \( \alpha = 0.05 \). Since \( p = 0.0418 \) is less than \( \alpha = 0.05 \), we reject the null hypothesis. Thus, there is sufficient evidence to conclude that customer satisfaction with home agents exceeds 80%.

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Key Concepts

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

Customer Satisfaction

Customer satisfaction is a key metric for businesses to gauge how happy customers are with their services or products. For Regional Airways, a satisfaction level above 80% is necessary to consider employing home agents. Satisfaction is often measured through surveys or feedback forms completed by customers. In this scenario, 252 out of 300 customers reported being satisfied. When businesses measure satisfaction:

• They understand customer needs better.
• It helps improve service quality by pinpointing weaknesses.
• Higher satisfaction rates are linked to loyalty and repeat business.

For Regional Airways, maintaining a high satisfaction rate ensures competitive advantage and customer retention.

Point Estimate

A point estimate provides an exact value to represent the characteristics of a population based on sample data. Here, it's the proportion of satisfied customers.To calculate the point estimate:

• Count the number of satisfied customers (252 in this case).
• Divide by the total number surveyed (300).

Mathematically, the point estimate is represented as:\[ \hat{p} = \frac{252}{300} = 0.84 \]This means that the point estimate for customer satisfaction is 84%. This value gives Regional Airways an initial understanding of how satisfied their trial customer base was with the home-based service agents.

Z-test

A z-test is a statistical method used to determine if there is a significant difference between sample and population means. In our context, it checks if customer satisfaction exceeds 80%.To set up a z-test for proportions:

• Define the null hypothesis (\( H_0: p \leq 0.80 \)) proposing satisfaction is 80% or less.
• Define the alternative hypothesis (\( H_a: p > 0.80 \)) proposing satisfaction is greater.

Using the point estimate (84%), the z-test formula is:\[ z = \frac{0.84 - 0.80}{\sqrt{\frac{0.80 \times 0.20}{300}}} \approx 1.73 \]This z-value helps determine how far the sample mean is from the hypothesized population mean in standard deviation terms.

P-value

The p-value quantifies the probability that the observed data would occur under the null hypothesis. If this value is low, it suggests that the sample provides enough evidence to reject the null hypothesis.Looking at our exercise:

• The calculated z-value is 1.73.
• The corresponding p-value is roughly 0.0418.

Since the p-value (0.0418) is less than the chosen level of significance (\( \alpha = 0.05 \)), we reject the null hypothesis. Thus, we have enough evidence to conclude that the home agents provide customer satisfaction exceeding 80%, which is the desired threshold for Regional Airways.