A bolt manufacturer is using a hypothesis test with α = 0.01 to see if the diameter of

their 0.75 cm diameter bolts are being manufactured properly. The goal is to have the
average bolt diameter be 0.75 ± 0.0075 cms.

To determine if the diameter of the bolts is being manufactured properly, the bolt manufacturer is using a hypothesis test with an alpha level of 0.01. The hypothesis being tested is whether the average bolt diameter is equal to 0.75 cm or not.

The null hypothesis, denoted as H0, is that the average bolt diameter is equal to 0.75 cm. The alternative hypothesis, denoted as Ha, is that the average bolt diameter is not equal to 0.75 cm.

Now, let's proceed with the steps to conduct the hypothesis test:

1. Determine the sample size (n) and calculate the mean and standard deviation of the sample. This will give you the sample average and the sample standard deviation of the bolt diameters.

2. Choose the appropriate statistical test. In this case, since the goal is to test whether the average bolt diameter is within a specific range, you can use a two-tailed t-test.

3. Determine the critical value(s) based on the alpha level (0.01) and the degrees of freedom (df), which is equal to n - 1. The critical value(s) can be found using a t-table or a statistical software.

4. Calculate the test statistic, which is the t-value. This can be calculated using the formula:

t = (sample mean - hypothesized mean) / (sample standard deviation / sqrt(n))

where the hypothesized mean is 0.75 cm, the sample mean is the calculated average bolt diameter, and the sample standard deviation is the calculated standard deviation of the bolt diameters.

5. Compare the calculated t-value with the critical value(s). If the calculated t-value falls within the critical region (i.e., beyond the critical value(s)), then you reject the null hypothesis. If the calculated t-value falls within the non-critical region (i.e., within the critical value(s)), then you fail to reject the null hypothesis.

6. Finally, interpret the results. If the null hypothesis is rejected, it means there is evidence to suggest that the average bolt diameter is not equal to 0.75 cm. If the null hypothesis is not rejected, it means there is insufficient evidence to suggest that the average bolt diameter is different from 0.75 cm.

Remember that this is just a general guideline for conducting a hypothesis test. It's important to consult a statistics textbook or a statistical software to ensure accuracy in the calculations and interpretation.