1. Think of one instance where an analyst would be interested in performing a hypothesis test about the population standard deviation $\sigma$. (p. 556)

2. What is the difference between $\sigma$ and ${\sigma }_0$? (p. 558)

3. Does it make sense to test whether $\sigma <0$? Explain. (p. 556)

4. What condition must be fulfilled for us to perform a hypothesis test about $\sigma$? (p. 556)

5. Explain how we can use a confidence interval to determine significance. (p. 561)

6. In the previous exercise, what must be the relationship between $\alpha$ and the confidence level? (p. 561)

1. State the hypotheses.
2. Find the ${\chi }^2$ critical values and state the rejection rule.
3. Calculate ${\chi }_{\text{data}}^2$.
4. Compare ${\chi }_{\text{data}}^2$ with the critical value or values. State the conclusion and interpretation.

7. Test whether $\sigma >1$, using $\alpha =0.05$. We have a sample of size $n=21$ and a sample variance of $s^2=3$.

8. Test whether $\sigma <5$, using $\alpha =0.05$. We have a sample of size $n=11$ and a sample variance of $s^2=25$.

9. Test whether $\sigma \ne 3$, using $\alpha =0.05$. We have a sample of size $n=16$ and a standard deviation of $s=2.5$.

10. Test whether $\sigma >10$, using $\alpha =0.01$. We have a sample of size $n=14$ and a standard deviation of $s=12$.

11. Test whether $\sigma <20$, using $\alpha =0.10$. We have a sample of size $n=8$ and a sample variance of $s^2=350$.

12. Test whether $\sigma \ne 5$, using $\alpha =0.05$. We have a sample of size $n=26$ with a standard deviation of $s=5$.

1. State the hypotheses and the rejection rule.
2. Calculate ${\chi }_{\text{data}}^2$ . Draw a ${\chi }^2$ distribution and indicate the location of ${\chi }_{\text{data}}^2$.
3. Find the $p$-value and indicate the $p$-value in your distribution in (b).
4. Compare the $p$-value with level of significance $\alpha =0.05$. State the conclusion and interpretation.

13. We are testing whether $\sigma >1$ and have a sample of size $n=21$ with a sample variance of $s^2=3$.

14. We are testing whether $\sigma <5$ and have a sample of size $n=11$ with a sample variance of $s^2=25$.

15. We are testing whether $\sigma \ne 3$ and have a sample of size $n=16$ with a standard deviation of $s=2.5$.

16. We are testing whether $\sigma >10$ and have a sample of size $n=14$ with a standard deviation of $s=12$.

17. We are testing whether $\sigma <20$, and have a sample of size $n=8$ and a sample variance of $s^2=350$.

18. We are testing whether $\sigma \ne 5$ and have a sample of size $n=26$ with a standard deviation of $s=5$.

19. A 95% ${\chi }^2$ confidence interval for $\sigma$ is (1, 4). Hypothesized values ${\sigma }_0$ are

20. A 99% ${\chi }^2$ confidence interval for $\sigma$ is (10, 25). Hypothesized values ${\sigma }_0$ are

21. A 90% ${\chi }^2$ confidence interval for $\sigma$ is (100, 200). Hypothesized values ${\sigma }_0$ are

22. A 95% ${\chi }^2$ confidence interval for $\sigma$ is (127, 698). Hypothesized values ${\sigma }_0$ are

23. Biomass Power Plants. Power plants around the country are retooling in order to consume biomass instead of or in addition to coal. The table contains a random sample of nine such power plants and the amount of energy generated in megawatts (MW) in 2014.¹⁶ The normality was checked in the Section 8.4 exercises. Test whether the population standard deviation differs from 25 MW, using level of significance $\alpha =0.05$.

24. Carbon Emissions. The following table represents the carbon emissions (in millions of tons) from consumption of fossil fuels, for a random sample of five nations.¹⁷ Test whether the population standard deviation is greater than 100 million tons, using level of significance $\alpha =0.10$.

25. Calories in Breakfast Cereals. A random sample of six well-known breakfast cereals yielded the following calorie data. Can we perform a ${\chi }^2$ test for the population standard deviation of the number of calories? Why or why not?

26. Deepwater Horizon Cleanup Costs. The following table represents the amount of money disbursed by BP to a random sample of six Florida counties, for cleanup of the Deepwater Horizon oil spill, in millions of dollars.¹⁸ The normality of the data was confirmed in the Section 8.1 exercises. Test whether the population standard deviation is less than $500,000, using level of significance $\alpha =0.05$.

27. Does Score Variability Differ by Gender? Recently, researchers have been examining the evidence for whether there is greater variability in boys' scores than girls' scores on cognitive abilities tests. For example, one study found that boys were overrepresented at both the top and the bottom of nonverbal reasoning tests and quantitative reasoning tests.¹⁹ Suppose that the standard deviation for girls' scores is known to be 50 points for a particular test and that the population of all scores is normal. A random sample of 101 boys has a sample variance of 2600. Test whether the population standard deviation for boys exceeds 50 points, using level of significance $\alpha =0.05$.

28. Heart Rate Variability. A reduction in heart rate variability is associated with elevated levels of stress, because the body continues to pump adrenaline after high-stress situations, even when at rest.²⁰ Suppose the standard deviation of heartbeats in the general population is 20 beats per minute, and that the population of heart rates is normal. A random sample of 50 individuals leading high-stress lives has a sample variance of 200 beats per minute. Test, using level of significance $\alpha =0.05$, whether the population standard deviation for those leading high-stress lives is lower than that in the general population.