Problem 24 - Basic Statistics for Business and Economics with Student

Question

Clark Heter is an industrial engineer at Lyons Products. He would like to determine whether there are more units produced on the night shift than on the day shift. Assume the population standard deviation for the number of units produced on the day shift is 21 and is 28 on the night shift. A sample of 54 day-shift workers showed that the mean number of units produced was 345\. A sample of 60 night-shift workers showed that the mean number of units produced was \(351 .\) At the .05 significance level, is the number of units produced on the night shift larger?

Step-by-Step Solution

Answer

The night shift produces significantly more units than the day shift at the 0.05 significance level.

Step 1: Define the Null and Alternative Hypotheses

The null hypothesis (H_0) states that the mean number of units produced on the night shift is equal to or less than the mean number of units produced on the day shift. The alternative hypothesis (H_1) states that the mean number of units produced on the night shift is greater than the mean on the day shift. Mathematically, we have:\[H_0: \mu_N \leq \mu_D \quad \text{and} \quad H_1: \mu_N > \mu_D\]where \( \mu_N \) is the mean of the night shift, and \( \mu_D \) is the mean of the day shift.

Step 2: Determine the Data Given

We are given the following data:- Day shift mean: \( \bar{x}_D = 345 \)- Night shift mean: \( \bar{x}_N = 351 \)- Population standard deviation (day): \( \sigma_D = 21 \)- Population standard deviation (night): \( \sigma_N = 28 \)- Day shift sample size: \( n_D = 54 \)- Night shift sample size: \( n_N = 60 \)- Significance level: \( \alpha = 0.05 \)

Key Concepts

Null Hypothesis

The null hypothesis is a foundational concept in hypothesis testing. When conducting a study, researchers propose a statement that there is no effect or no difference. For Clark Heter's exercise, the null hypothesis (denoted as \( H_0 \)) is that the mean number of units produced on the night shift is equal to or less than those produced on the day shift. In simpler terms, it suggests nothing special is happening—night and day shifts produce about the same number of units or the day shift produces more.

This hypothesis assumes that any observed difference is due to random chance rather than an actual underlying difference in productivity. By assuming no difference initially, researchers can test to see if their evidence is strong enough to reject this assumption based on statistical results.

The mathematical expression for this hypothesis would be \( H_0: \mu_N \leq \mu_D \), where \( \mu_N \) represents the mean number of units produced on the night shift, and \( \mu_D \) represents the mean number of units on the day shift.

Alternative Hypothesis

While the null hypothesis assumes no difference, the alternative hypothesis suggests the opposite. It proposes that there is a significant effect or difference. For the Lyons Products case, the alternative hypothesis (denoted as \( H_1 \)) states that the mean number of units produced on the night shift is greater than that on the day shift. This hypothesis is an assertion that there is something noteworthy happening—indicative of higher productivity on the night shift.

Formally, the alternative hypothesis is expressed as \( H_1: \mu_N > \mu_D \)—asserting night shifts produce more than day shifts. If the data is strong enough to support this, Clark can conclude that night shifts indeed yield more units. Researchers prefer the alternative to be directional whenever prior information or business interests are at stake, as seen in this exercise.

Significance Level

The significance level, often represented by \( \alpha \), is a threshold used by researchers to decide whether to reject the null hypothesis. It represents the probability of rejecting the null hypothesis when it is actually true. In Clark's analysis, a significance level of 0.05 was chosen, which is common in many research studies, indicating a 5% risk of concluding a difference when there is none.

Setting a significance level helps balance between Type I error (false positive) and Type II error (false negative). For instance, if the significance level is set too low, you might miss a true effect due to stringent criteria. Conversely, a high level might falsely detect an effect that doesn't exist.

With \( \alpha = 0.05 \), Clark Heter has established a confidence level of 95% that his findings aren't merely due to random chance. His task now is to use this threshold to evaluate the evidence and decide whether the observed difference in production is statistically significant.