Deck 9: Inferences About Variances

Question 1

The following summary statistics are available for two dependent random samples:

s₁² = 36, s₂² = 16, n = 22, r₁₂ = .70.

Test the following hypotheses at the .05 level of significance:

$\begin{array}{l}H_{0}: \sigma_{1}^{2}-\sigma_{2}^{2}=0 \\H_{1}: \sigma_{1}^{2}-\sigma_{2}^{2} \neq 0\end{array}$

Answer 1

The degrees of freedom (v) are equal to n - 2 = 22 - 2 = 20. The standard deviations are, respectively, 6 and 4 (i.e., the square root of the variance terms provided).

$t=\frac{s_{1}^{2}-s_{2}^{2}}{2 s_{1} s_{2} \sqrt{\frac{1-r_{12}^{2}}{v}}}=\frac{36-16}{2(6) 4 \sqrt{\frac{1-(70)^{2}}{20}}}=\frac{24}{1.224}=19.608$
_{From Appendix Table 2, and using an alpha level of .05 and degrees of freedom (v) of 20, we determine the critical values to be +/? 2.086. As the test statistic (19.608) exceeds the critical value, our decision is to reject the null hypothesis.}

Question 2

The following summary statistics are available for two dependent random samples:

s₁² = 100, s₂² = 225, n = 122, r₁₂ = .50.

Test the following hypotheses at the .05 level of significance:

$\begin{array}{l}H_{0}: \sigma_{1}^{2}-\sigma_{2}^{2}=0 \\H_{1}: \sigma_{1}^{2}-\sigma_{2}^{2} \neq 0\end{array}$

Answer 2

The degrees of freedom (v) are equal to n - 2 = 122 - 2 = 120. The standard deviations are, respectively, 10 and 15 (i.e., the square root of the variance terms provided).

$t=\frac{s_{1}^{2}-s_{2}^{2}}{2 s_{1} s_{2} \sqrt{\frac{1-r_{12}^{2}}{v}}}=\frac{100-225}{2(10) 15 \sqrt{\frac{1-(50)^{2}}{120}}}=\frac{-115}{13.693}=-8.398$
_{From Appendix Table 2, and using an alpha level of .05 and degrees of freedom (v) of 120, we determine the critical values to be +/- 1.98. As the test statistic
(-8.398) exceeds the critical value, our decision is to reject the null hypothesis.}

Question 3

A random sample of 51scores is collected with a sample mean of 75 and a sample variance of 10. Test the following hypotheses at the .05 level of significance:

$\begin{array}{l}H_{0}: \sigma^{2}=7 \\H_{1}: \sigma^{2} \neq 7\end{array}$

Answer 3

Degrees of freedom (_v) are equal to _{n - 1, thus 51 - 1 = 50.
The hypothesized population variance is provided to us (7).
Plugging these values into the formula, we find the test statistic value:}
$\chi^{2}=\frac{v s^{2}}{\sigma_{0}^{2}}=\frac{50(10)}{7}=71.429$
_{From Appendix Table 3, and using an alpha level of .05 (i.e., tabled values listed under .975 and .025) and degrees of freedom (v) of 50, we determine the critical values to be 71.4202 and 32.3574. As the test statistic (71.429) does exceeds one of the critical values by falling into the upper-tail critical region (71.429 > 71.4202), our decision is to reject the null hypothesis.}

Question 4

A random sample of 9 scores is collected with a sample mean of 62 and a sample variance of 16. Test the following hypotheses at the .05 level of significance:

$\begin{array}{l}H_{0}: \sigma^{2}=15 \\H_{1}: \sigma^{2} \neq 15\end{array}$

Question 5

A random sample of 5 scores is collected with a sample mean of 8 and a sample variance of 9. Test the following hypotheses at the .05 level of significance:

$\begin{array}{l}H_{0}: \sigma^{2}=5 \\H_{1}: \sigma^{2} \neq 5\end{array}$

Question 6

In what is a researcher interested when conducting a test of inference for independent variances?

A) Whether the population variance for one group is different than the population variance for one or more other independent groups
B) Whether the population variance for one group is different than the population variance for one or more other dependent groups
C) Whether the population mean difference for one group is different than a hypothesized value
D) Whether the sample mean of group 1 differs from the sample mean of group 2
E) Whether the population standard deviation is within a ratio of 1:4
F) Whether two groups have the same population mean and variance

Question 7

Which of the following are examples of types of research questions that can be answered with a test of inference about independent variances? Select all that apply.

A) Does the variation in the time to complete a task different for subjects in a treatment group as compared to subjects in a control group?
B) Is variation in height different for children who drink the recommended number of glasses of water per day as compare to children who do not?
C) Is there a relationship between the variation of BMI in adults and weight in middle school?
D) Are there similar proportions of public institutions who declined in federal spending in 2000 as compared to 2010?

Question 8

Which of the following are examples of types of research questions that can be answered with a test of inference about independent variances?

A) Is there a relationship between book price and tuition?
B) Are there similar proportions of U.S. versus non-U.S. board graduates?
C) Is the variation in graduation rate for public colleges different than the graduation rate for private colleges?
D) Does weight vary for athletes who train 6 hours per day as compared to athlete who train 8 hours per day?
E) Is there a relationship between satisfaction with a job's salary and satisfaction with a job's level of responsibility?
F) Is there a mean difference in engagement in the workplace based on whether an employee is full-time or part-time?