Production Manager Of MPS Audio Systems Inc Is Concerned
The Production Manager Of Mps Audio Systems Inc Is Concerned About Th
The production manager of MPS Audio Systems Inc. is concerned about the idle time of workers. In particular, he would like to know if there is a difference in the idle minutes for workers on the day shift and the evening shift. The information below is the number of idle minutes yesterday for the five day-shift workers and the six evening-shift workers. State the decision rule. Use the .05 significance level. H₀: Idle minutes are the same. H₁: Idle minutes are not the same. (Negative amount should be indicated by a minus sign. Round your answers to two decimal places.) Reject H₀ if z
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The primary concern of the production manager at MPS Audio Systems Inc. is to determine whether there is a statistically significant difference in the idle minutes between workers on the day shift and those on the evening shift. To explore this, a hypothesis test for the difference between two means is appropriate. The significance level for this test is set at 0.05, which provides a standard criterion for making a decision about the null hypothesis based on the calculated test statistic and the critical value.
The null hypothesis (H₀) states that there is no difference in the average idle minutes between the two shifts: H₀: μ₁ = μ₂. Conversely, the alternative hypothesis (H₁) suggests that there is a difference: H₁: μ₁ ≠ μ₂. This is a two-tailed test because the concern is whether the means differ in either direction.
The procedure involves gathering the sample data—idle minutes for five day-shift workers and six evening-shift workers—and calculating the sample means and standard deviations for each group. Next, the pooled standard deviation and the test statistic, a z-score, are computed. The z-score formula for the difference between two means with known variances (or assuming large samples) is:
z = (x̄₁ - x̄₂) / √(σ₁²/n₁ + σ₂²/n₂)
where x̄₁ and x̄₂ are the sample means, σ₁ and σ₂ are the population standard deviations or approximations based on sample standard deviations, and n₁ and n₂ are the respective sample sizes.
For this particular test, with the calculated z-value, the decision rule at a two-tailed significance level of 0.05 is to reject the null hypothesis if the z-score falls into the critical region, which is z 1.96, based on standard normal distribution tables. Our focus here is primarily on the left tail (z
In conclusion, if the calculated z-value is less than -1.96, we reject H₀, indicating there is statistically significant evidence at the 0.05 level that the idle minutes differ between the day and evening shifts. Conversely, if z > -1.96, we fail to reject H₀, suggesting no significant difference exists.
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The analysis of worker idle time at MPS Audio Systems Inc. aims to address whether there is a meaningful difference between the shifts. This kind of hypothesis testing is crucial in industrial and organizational settings to optimize productivity and resource allocation. Specifically, the question centers on whether the observed variations in idle times are statistically significant or attributable to random chance.
To conduct this hypothesis test, we start with formulating the null hypothesis (H₀) and the alternative hypothesis (H₁). The null hypothesis assumes that the means of idle minutes for day and evening shifts are equal, indicating no difference in productivity or idle time. The alternative hypothesis posits a difference, which could be in either direction as the test is two-tailed.
Gathering data involves calculating sample means and standard deviations for each shift using the provided idle minutes. Assuming the data give us five observations from the day shift workers and six from the evening shift workers, the next step is to compute the sample statistics:
- Sample mean of day shift: x̄₁
- Sample mean of evening shift: x̄₂
- Sample standard deviations: s₁, s₂
Using these, the standard error of the difference between means is calculated, and the z-test statistic is derived accordingly:
z = (x̄₁ - x̄₂) / √(s₁²/n₁ + s₂²/n₂)
The decision rule hinges on the critical z-value for a significance level of 0.05 in a two-tailed test, which is ±1.96. Given the problem's focus on the left tail, the rejection criterion is if z
If the calculated z falls into the critical region (z
This analysis helps management make informed decisions regarding process improvements, staffing, and scheduling. Recognizing whether shift differences in idle time are statistically significant can point to underlying operational inefficiencies or areas needing attention, aiding in targeted interventions to enhance productivity.
Overall, applying proper statistical tests with significance level considerations allows for a data-driven approach to managing human resource efficiency in manufacturing environments like MPS Audio Systems Inc.
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