To test the claim that the standard deviation of the population of pennies is greater than 0.01000 g, we can use a chi-square test for the population variance (which is related to the standard deviation).

Step-by-step process:

1. Define the null and alternative hypotheses:

   • $H_0$ (null hypothesis): The population standard deviation is less than or equal to 0.01000 g. $H_0: \sigma \leq 0.01000$
   • $H_1$ (alternative hypothesis): The population standard deviation is greater than 0.01000 g. $H_1: \sigma > 0.01000$

2. Test statistic: The chi-square test statistic for variance is given by: $\chi^2 = \frac{(n - 1) \cdot s^2}{\sigma_0^2}$ Where:

   • $n$ is the sample size.
   • $s$ is the sample standard deviation.
   • $\sigma_0$ is the hypothesized population standard deviation (0.01000 g).

   Plugging in the values:

   • $n = 37$
   • $s = 0.01648 \, \text{g}$
   • $\sigma_0 = 0.01000 \, \text{g}$

   First, calculate $s^2$ and $\sigma_0^2$: $s^2 = (0.01648)^2 = 0.000271635 \quad \text{and} \quad \sigma_0^2 = (0.01000)^2 = 0.00010000$

   Now, compute the chi-square statistic: $\chi^2 = \frac{(37 - 1) \cdot 0.000271635}{0.00010000} = \frac{36 \cdot 0.000271635}{0.00010000} = 97.7886$

3. Degrees of freedom (df): The degrees of freedom for the chi-square test are $n - 1$. So: $df = 37 - 1 = 36$

4. Find the critical value: At a significance level of $\alpha = 0.05$ for a right-tailed test with 36 degrees of freedom, we need to find the critical value from the chi-square distribution table.

   The critical value for $\chi^2_{0.05, 36}$ is approximately 50.998 (based on chi-square distribution tables).

5. Decision rule:

   • If $\chi^2$ (calculated) > $\chi^2_{\text{critical}}$, reject the null hypothesis.
   • If $\chi^2$ (calculated) ≤ $\chi^2_{\text{critical}}$, fail to reject the null hypothesis.

6. Conclusion: Since the calculated $\chi^2 = 97.7886$ is greater than the critical value $50.998$, we reject the null hypothesis.

Final conclusion:

There is sufficient evidence at the 0.05 significance level to support the claim that the population standard deviation is greater than 0.01000 g.

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Do you want more details or have any questions?
Here are 5 related questions to expand on this concept:

1. How would the hypothesis test change if the sample size were larger?
2. What is the effect of choosing a different significance level (e.g., 0.01) on the test result?
3. How can we compute a confidence interval for the population variance?
4. Why is the chi-square distribution used for variance testing?
5. What are the assumptions behind using the chi-square test for variance?

Tip: The chi-square test for variance is sensitive to non-normal data, so it's important to verify the normality assumption of your sample!