A manufacturer has received complaints that aging production equipment is forcing workers to work overtime in order to meet production quotas. Historically, the average hours worked per week at the company is normally distributed, with a mean of 40 hours and a standard deviation of 7.5 hours. If a sample of 30 workers found an average work week of 45 hours, conduct a hypothesis test to determine if there is significant evidence (at a 5 percent level of significance) that the length of the average work week has increased.

Question 2
A human resources manager is assessing the popularity of the current compensation plan. The manager collects two samples of 50 employees each from two different divisions in the company. In the first sample, 40 individuals indicated that they are in favor of the current plan. In the second sample, only 30 individuals indicated that they are in favor of the current plan. Conduct a hypothesis test at a 5 percent level of significance to determine whether the proportion in favor of the plan is significantly different between the two samples

To conduct a hypothesis test for the given scenarios, we will follow the general steps:

Step 1: Define the Null and Alternative Hypotheses:
- Null Hypothesis (H0): There is no significant evidence of change/ difference.
- Alternative Hypothesis (Ha): There is significant evidence of change/ difference.

Step 2: Determine the Level of Significance (α):
The level of significance dictates the risk of making a Type I error (rejecting the null hypothesis when it is true). In both scenarios, α is given as 5% (0.05).

Step 3: Choose the Appropriate Statistical Test:
- Scenario 1: Comparing the means of two independent samples -> Two-Sample t-test
- Scenario 2: Comparing the proportions of two independent samples -> Two-Sample Proportion Test

Now let's proceed with each scenario:

Scenario 1: Hypothesis test for average work week duration

Given:
- Population mean (μ) = 40 hours
- Population standard deviation (σ) = 7.5 hours
- Sample size (n) = 30 workers
- Sample mean (x̄) = 45 hours
- α (Level of significance) = 0.05

Step 4: Compute the test statistic and p-value using the appropriate formula:
For a two-sample t-test, we calculate the t-statistic using the formula:
t = (x̄ - μ) / (s / √n)
where s is the sample standard deviation, and n is the sample size.

Next, we find the degrees of freedom (df), which is the sum of the sample size minus 2 for two independent samples.

With the calculated t-statistic and df, we can find the p-value either using statistical software or lookup tables.

Step 5: Make a Decision:
If the p-value is less than the level of significance (α), we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.

Scenario 2: Hypothesis test for the difference in proportions

Given:
- Sample size for the first sample (n1) = 50 employees
- Sample size for the second sample (n2) = 50 employees
- Number in favor of the current plan in the first sample (x1) = 40 employees
- Number in favor of the current plan in the second sample (x2) = 30 employees
- α (Level of significance) = 0.05

Step 4: Compute the test statistic and p-value using the appropriate formula:
For a two-sample proportion test, we calculate the test statistic using the formula:
z = (p1 - p2) / √[(p * (1 - p)) * ((1/n1) + (1/n2))],
where p1 and p2 are the sample proportions, and p is the pooled proportion.

Next, we find the p-value either using statistical software or lookup tables.

Step 5: Make a Decision:
If the p-value is less than the level of significance (α), we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.

Remember that hypothesis tests provide evidence, but they do not prove or disprove a claim with absolute certainty.