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LOS i: Identify the appropriate test statistic and interpret the results for a hypothesis test concerning 1) the variance of a normally distributed population, and 2) the equality of the variances of two normally distributed populations based on two independent random samples.

Which of the following statements about the variance of a normally distributed population is least accurate?

A)	The Chi-squared distribution is a symmetric distribution.

B)	The test of whether the population variance equals σ₀²requires the use of a Chi-squared distributed test statistic, [(n ? 1)s²] / σ₀².

C)	A test of whether the variance of a normally distributed population is equal to some value σ₀², the hypotheses are: H₀: σ² = σ₀², versus H_a: σ² ≠ σ₀².

The Chi-squared distribution is not symmetrical, which means that the critical values will not be numerically equidistant from the center of the distribution, though the probability on either side of the critical values will be equal (that is, if there is a 5% level of significance and a two-sided test, 2.5% will lie outside each of the two critical values).

A test of the population variance is equal to a hypothesized value requires the use of a test statistic that is:

In tests of whether the variance of a population equals a particular value, the chi-squared test statistic is appropriate.

A munitions manufacturer claims that the standard deviation of the powder packed in its shotgun shells is 0.1% of the stated nominal amount of powder. A sport clay association has reviewed a sample of 51 shotgun shells and found a standard deviation of 0.12%. What is the Chi-squared value, and what are the critical values at a 95% confidence level, respectively?

To compare standard deviations we use a Chi-square statistic. X² = (n – 1)s² / σ₀² = 50(0.0144) / 0.01 = 72. With 50 df, the critical values at the 95% confidence level are 32.357 and 71.420. Since the Chi-squared value is outside this range, we can reject the hypothesis that the standard deviations are the same.

The variance of 100 daily stock returns for Stock A is 0.0078. The variance of 90 daily stock returns for Stock B is 0.0083. Using a 5% level of significance, the critical value for this test is 1.61. The most appropriate conclusion regarding whether the variance of Stock A is different from the variance of Stock B is that the:

A)	variance of Stock B is significantly greater than the variance of Stock A.

B)	variances are not equal.

C)	variances are equal.

A test of the equality of variances requires an F-statistic. The calculated F-statistic is 0.0083/0.0078 = 1.064. Since the calculated F value of 1.064 is less than the critical F value of 1.61, we cannot reject the null hypothesis that the variances of the 2 stocks are equal.

In order to test if Stock A is more volatile than Stock B, prices of both stocks are observed to construct the sample variance of the two stocks. The appropriate test statistics to carry out the test is the:

The F test is used to test the differences of variance between two samples.

Abby Ness is an analyst for a firm that specializes in evaluating firms involved in mineral extraction. Ness believes that the earnings of copper extracting firms are more volatile than those of bauxite extraction firms. In order to test this, Ness examines the volatility of returns for 31 copper firms and 25 bauxite firms. The standard deviation of earnings for copper firms was $2.69, while the standard deviation of earnings for bauxite firms was $2.92. Ness’s Null Hypothesis is σ₁² = σ₂². Based on the samples, can we reject the null hypothesis at a 95% confidence level using an F-statistic and why? Null is:

A)	rejected. The F-value exceeds the critical value by 0.849.

B)	rejected. The F-value exceeds the critical value by 0.71.

C)	not rejected. The critical value exceeds the F-value by 0.71.

F = s₁² / s₂² = $2.92² / $2.69² = 1.18

From an F table, the critical value with numerator df = 24 and denominator df = 30 is 1.89.

The use of the F-distributed test statistic, F = s₁² / s₂², to compare the variances of two populations does NOT require which of the following?

A)	two samples are of the same size.

B)	populations are normally distributed.

C)	samples are independent of one another.

The F-statistic can be computed using samples of different sizes. That is, n₁ need not be equal to n₂.

The test of the equality of the variances of two normally distributed populations requires the use of a test statistic that is:

The F-distributed test statistic, F = s₁² / s₂², is used to compare the variances of two populations.