Q: Given a sample variance $$s^2 = 1225$, sample size n = 20$, and hypothesized variance $$\$$sigma^2 = 900$, what is the test statistic for a chi-square test?

$$\$$chi^2$$ = \frac{19 \times 1225}{900}$$

Question 1

Which of the following best describes variance in a data set?

Accepted Answer

A measurement of how far each number in a data set is from the mean (average), and thus from every other number in the set.

Question 2

Why do we use variance ($$\$$sigma^2$$) instead of standard deviation ($$\sigma$$) in hypothesis testing for population spread?

Accepted Answer

Because there is no statistical test that directly tests the standard deviation ($$\sigma).$$

Question 3

A company wants to test if the variance of job completion times has increased from $$900$$ minutes$$^2. A $$sample of $$n=20$$ jobs has $$s=35$$ minutes. What is the correct null hypothesis ($$H_0$$)?

Accepted Answer

$$H_0$$: \$$sigma^2$$ \leq 900$$

Question 4

Given a sample variance $$s^2 = 1225$, sample size n = 20$, and hypothesized variance $$\$$sigma^2 = 900$, what is the test statistic for a chi-square test?

Accepted Answer

$$\$$chi^2$$ = \frac{19 	imes 1225}{900}$$

Question 5

If the calculated chi-square test statistic is $$25.86$$ and the critical value at $$\alpha = 0.10$$ with $$19$$ degrees of freedom is $$27.20$$, what is the correct decision?

Accepted Answer

Fail to reject the null hypothesis.

Question 6

Which distribution is used for hypothesis testing of a single population variance?

Accepted Answer

Chi-square distribution

Question 7

A 95% confidence interval for a population variance is calculated as $$783.06 \leq \$$sigma^2$$ \leq 3867.09. $$What does this mean?

Accepted Answer

With 95% confidence, the true population variance is between $$783.06$$ and $$3867.09.$$

Question 8

Which formula gives the confidence interval for a population variance $$\$$sigma^2$$ based on a sample variance s^2$ and degrees of freedom df$?

Accepted Answer

$$\left( \frac{(n-1)s^2$}{\$$chi^2$$_{U, \alpha/2}}, \frac{(n-1)s^2$}{\$$chi^2$$_{L, \alpha/2}} \right)$$

Question 9

Which of the following is NOT an assumption for using the F-test to compare two population variances?

Accepted Answer

Sample sizes must be equal.

Question 10

A fast food chain tests two drive-thru systems. System 1: $$n=20$$, $$s_1=45$ sec; System 2: n=24$, s_2$=67 sec. $$What is the F test statistic to compare variances?

Accepted Answer

$$F = \frac{4489}{2025}$$

Question 11

If the F test statistic is $$2.22$$ and the critical value at $$\alpha = 0.05 is$$ $$2.465$$, what is the correct conclusion?

Accepted Answer

Fail to reject the null hypothesis; no evidence of difference in variances.

Question 12

Which of the following is the correct formula for the chi-square test statistic for variance?

Accepted Answer

$$\$$chi^2$$ = \frac{(n-1)s^2$}{\$$sigma^2$$}$$

Question 13

In a two-tailed test for variance, when do you reject the null hypothesis?

Accepted Answer

If $$\$$chi^2$$_{test}  \$$chi^2$$_{U, \alpha/2}$$

Question 14

A sample of 14 Saturday night sales at a theater has $$s = 38.6$$, $$n = 14. $$What are the degrees of freedom for the confidence interval for variance?

Accepted Answer

13

Question 15

Which Excel function would you use to find the right-tail critical value for a chi-square test with $$\alpha = 0.1$$ and $$df = 19$$?

Accepted Answer

=CHISQ.INV.RT(0.1, 19)

Question 16

A business wants to compare the variability in sales between two cities using the F-test. What is the correct order for the numerator and denominator in the F statistic?

Accepted Answer

Larger sample variance in the numerator, smaller in the denominator.

Question 17

Which of the following is a correct interpretation of a 95% confidence interval for standard deviation $$27.98 \leq \sigma \leq 62.19$$?

Accepted Answer

With 95% confidence, the true population standard deviation is between $$27.98$$ and $$62.19.$$

Hypothesis Tests and Estimation for Population Variance: Chi-Square and F-Tests

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