One Clark Heter Is An Industrial Engineer At Lyons Products

One Clark Heter Is An Industrial Engineer At Lyons Products He Woul

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. A sample of 50 day-shift workers showed that the mean number of units produced was 353, with a population standard deviation of 25. A sample of 55 night-shift workers showed that the mean number of units produced was 363, with a population standard deviation of 31 units. At the 0.01 significance level, is the number of units produced on the night shift larger? This is a one-tailed test. The decision rule is to reject the null hypothesis if the Z value is less than a certain critical value. The test statistic is calculated using the formula for the Z-test for two means with known standard deviations.

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The primary objective of this analysis is to determine whether the night shift employees at Lyons Products produce significantly more units than their day shift counterparts, leveraging a hypothesis testing framework. The question posed is statistically tested with a sample size of 50 day-shift workers and 55 night-shift workers, with known population standard deviations. This setup indicates the use of a Z-test for the comparison of two means with known variances, suitable when the sample size is sufficiently large, and the population standard deviations are specified.

First, formal hypotheses are established: the null hypothesis (H0) assumes no difference or that the day shift's mean output is at least equal to or greater than the night shift's mean, which can be formulated as H0: μ_day ≥ μ_night; the alternative hypothesis (H1) suggests the night shift produces more units, expressed as H1: μ_night > μ_day. Because we are testing if the night shift produces more units, this is a one-tailed right test. Significance level (α) is set at 0.01, which indicates a stringent criterion for rejecting the null hypothesis, reducing the likelihood of Type I errors.

The formula for the test statistic Z in the context of known variances is:

Z = (x̄_night - x̄_day) / √( (σ_night² / n_night) + (σ_day² / n_day) )

where x̄ represents the sample mean, σ the population standard deviation, and n the sample size.

Plugging in the provided data:

x̄_day = 353, σ_day = 25, n_day = 50

x̄_night = 363, σ_night = 31, n_night = 55

Calculating the denominator:

√( (31² / 55) + (25² / 50) ) = √( (961 / 55) + (625 / 50) ) = √(17.47 + 12.5) = √(29.97) ≈ 5.48

Calculating the numerator:

363 - 353 = 10

The test statistic:

Z = 10 / 5.48 ≈ 1.82

Next, the critical value for a one-tailed test at a significance level of 0.01 is approximately 2.33 (from standard normal distribution tables). Since the calculated Z-value (≈1.82) is less than 2.33, we fail to reject the null hypothesis. This suggests that, based on the sample data, there is not enough evidence at the 1% significance level to conclude that the night shift workers produce more units than the day shift workers.

In conclusion, while the observed difference in sample means favors higher productivity on the night shift, the difference is not statistically significant under the chosen significance level. Therefore, we cannot claim a definitive increase in productivity during the night shift based on this data. However, further studies with larger samples or different methodologies might provide more insight into production differences across shifts.

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