This problem involves hypothesis testing, specifically comparing a test statistic to critical values to make a decision about rejecting or failing to reject the null hypothesis $H_0$.

Let's break it down:

1. Null Hypothesis ($H_0$): The standard deviation of the errors is not greater than $32.2$ ft ($\sigma \leq 32.2$).
2. Alternative Hypothesis ($H_a$): The standard deviation of the errors is greater than $32.2$ ft ($\sigma > 32.2$).

This is a right-tailed test since the claim focuses on whether the standard deviation is greater.

Steps:

1. Compare Test Statistic to Critical Value:

   • If the test statistic is greater than the critical value, reject $H_0$.
   • If the test statistic is less than or equal to the critical value, fail to reject $H_0$.

2. Conclusion:

   • If $H_0$ is rejected, there is sufficient evidence to support the claim.
   • If $H_0$ is not rejected, there is insufficient evidence to support the claim.

Fill in the Blanks:

1. Depending on the test statistic's relationship to the critical value:
   • "Since the test statistic is greater than the critical value, reject $H_0$."
     • There is sufficient evidence to support the claim that the new production method has errors with a standard deviation greater than $32.2$ ft.
   • "Since the test statistic is less than the critical value, fail to reject $H_0$."
     • There is insufficient evidence to support the claim that the new production method has errors with a standard deviation greater than $32.2$ ft.

Clarifications:

• The exact conclusion depends on the numerical values for the test statistic and critical value, which you should calculate or be provided.
• Let me know if you need help computing or interpreting the test statistic or critical value.

Would you like to go over calculations or specific examples?

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5 Questions to Expand Understanding:

1. How do you calculate the test statistic for a variance or standard deviation hypothesis test?
2. What is the formula for finding critical values in a chi-square test?
3. How does the significance level ($\alpha$) affect the decision rule?
4. Can you provide an example where $H_0$ is not rejected, and why?
5. What assumptions must be satisfied for this hypothesis test to be valid?

Tip:

When dealing with hypothesis tests for standard deviation or variance, always ensure that the sample data is from a normally distributed population. If this assumption is violated, the test may not yield accurate results.