Question 1 Production Rates

A manufacturing company wishes to find out whether the productivity of its workforce has increased now that they have a new machine. The factory examined the records for a random sample of 8 hours over the past month. The hourly production rates for these 8 hours were:
142 175 162 158 190 154 160 185
Suppose the production rate before the use of the new machine is 150 units per hour. Test at 5% if the new machine has increased workers’ productivity on average using
(a) p-value approach, and
(b) critical value approach.

hello hi did you figure out how to do this? ahahha help

To test if the new machine has increased workers' productivity on average, we can perform a hypothesis test. Let's go through the steps for both the p-value approach and the critical value approach.

(a) P-value approach:
Step 1: State the null and alternative hypotheses:
- Null hypothesis (H0): The new machine has not increased workers' productivity on average. The mean production rate is still 150 units per hour.
- Alternative hypothesis (Ha): The new machine has increased workers' productivity on average. The mean production rate is greater than 150 units per hour.

Step 2: Determine the test statistic and the distribution:
Since the sample size is small (n = 8) and the population standard deviation is unknown, we will use a t-test. The test statistic in this case is the t-statistic.

Step 3: Calculate the test statistic:
We need to calculate the sample mean and the standard error.

Sample Mean (x̄) = (142 + 175 + 162 + 158 + 190 + 154 + 160 + 185) / 8 = 160.125
Standard Error (SE) = sample standard deviation / sqrt(n)
First, calculate the sample standard deviation:
Sample standard deviation = sqrt((Σ(xi - x̄)^2) / (n - 1))
= sqrt((142 - 160.125)^2 + (175 - 160.125)^2 + ... + (185 - 160.125)^2) / (8 - 1)
= sqrt(1405.75) / 7
≈ 7.52
So, SE = 7.52 / √8 ≈ 2.66

Now, calculate the t-statistic:
t = (x̄ - μ) / (SE / √n)
where μ is the hypothesized mean (150), x̄ is the sample mean (160.125), SE is the standard error (2.66), and n is the sample size (8).
t = (160.125 - 150) / (2.66 / √8)
≈ 3.79

Step 4: Determine the p-value:
Calculate the p-value using the t-distribution with n - 1 degrees of freedom (8 - 1 = 7).
In this case, since we have a one-tailed test (we are testing if the mean is greater than 150), we need to find the probability of t being greater than 3.79.
Using a t-table or statistical software, we find that the p-value is approximately 0.004.

Step 5: Decide:
Since the p-value (0.004) is less than the significance level (5%), we reject the null hypothesis. We have evidence to suggest that the new machine has increased workers' productivity on average.

(b) Critical value approach:
Step 1: State the null and alternative hypotheses (same as in the p-value approach).

Step 2: Determine the test statistic and the distribution (same as in the p-value approach).

Step 3: Calculate the test statistic (same as in the p-value approach).

Step 4: Determine the critical value:
For a significance level of 5%, the critical value can be found using the t-distribution with n - 1 degrees of freedom (7).
Using a t-table or statistical software, we find the critical value to be approximately 1.895 for a one-tailed test (testing if the mean is greater than 150).

Step 5: Decide:
If the test statistic is greater than the critical value (3.79 > 1.895), we reject the null hypothesis. We have evidence to suggest that the new machine has increased workers' productivity on average.

In both the p-value approach and the critical value approach, we came to the conclusion that the new machine has increased workers' productivity on average.