Suppose That In Example 11.1 We Wanted To Determine Whether

Suppose that in Example 11.1 we wanted to determine whether there was sufficient evidence to conclude that the new system would not be costeffective. Set up the null and alternative hypotheses and discuss the consequences of Type I and Type II errors.

This assignment focuses on formulating hypotheses and conducting a hypothesis test within the context of evaluating the cost-effectiveness of a new billing system at a department store, as illustrated in Example 11.1. The primary objective is to determine whether there is sufficient statistical evidence to conclude that the new billing system is not cost-effective, based on the mean monthly accounts. The task involves setting up null and alternative hypotheses, discussing the implications of Type I and Type II errors, executing the appropriate test, and comparing the conclusion with the original example to understand the decision-making process in hypothesis testing.

Paper For Above instruction

In the realm of managerial decision-making, hypothesis testing serves as an essential statistical tool to inform whether new systems, such as billing processes, are justified economically. Building upon Example 11.1, where a department store's manager considers implementing a new billing system, this paper explores the process of testing whether the system is cost-effective through formal statistical hypotheses. The core question is: does the data provide sufficient evidence to reject the notion that the true mean monthly account is less than or equal to a threshold of $170, and thus, support the claim that the system is financially beneficial?

First, we define the null and alternative hypotheses. The null hypothesis (H₀) represents the assumption of no effect or no difference—in this case, that the new system is not cost-effective, which can be operationalized as the population mean monthly account being less than or equal to $170. Mathematically, H₀: μ ≤ $170. Conversely, the alternative hypothesis (H₁) reflects the research hypothesis that the system is indeed cost-effective, implying that the average monthly account exceeds $170. Therefore, H₁: μ > $170. This setup aligns with a one-tailed test aimed at detecting whether the mean exceeds a specific threshold, indicating cost-effectiveness.

Next, the implications of Type I and Type II errors are critical to decision-making. A Type I error occurs if we reject H₀ when it is true—falsely concluding that the system is cost-effective when it is not. This may lead to unnecessary implementation costs and potential financial losses. A Type II error happens if we fail to reject H₀ when H₁ is true—failing to recognize the system's benefit when it actually is cost-effective, potentially resulting in missed profit opportunities.

Applying the data from the example, the sample mean account value is $178, with a sample size of 400. The known population standard deviation is $65. To test the hypothesis, we employ a z-test given the large sample size. The test statistic is computed as:

z = (x̄ - μ₀) / (σ / √n) = (178 - 170) / (65 / √400) = (8) / (65 / 20) = 8 / 3.25 ≈ 2.46

Using a significance level of α = 0.05, the critical value for a right-tailed test is approximately 1.645. Because the calculated z-value (2.46) exceeds 1.645, we reject the null hypothesis. The statistical evidence suggests that the mean monthly account exceeds $170, supporting the conclusion that the new billing system is cost-effective. This aligns with the conclusion in Example 11.1, reaffirming the initial inference.

However, if the test had resulted in a z-value less than 1.645, we would fail to reject H₀, indicating insufficient evidence to conclude cost-effectiveness. It is crucial to recognize that failing to reject H₀ does not prove that μ ≤ $170; rather, it suggests that the data do not provide strong enough evidence to affirm the alternative hypothesis at the specified significance level.

In summary, constructing precise hypotheses, understanding the consequences of potential errors, and correctly executing the test are essential steps in data-driven managerial decisions. The example demonstrates that with sufficient evidence, the store manager can confidently conclude that the new billing system will be beneficial, thus facilitating informed and rational decision-making processes.

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