Question

The following data are drawn from a normal population: (92, 93, 54, 58,74, 53, 63, 83, 64, 51). At the 1% significant level, test whether there isenough evidence to conclude that the population variance is less than 480.b. rhe another sample is also drawn from a normal population which is independent of the first one, and data are: (78, 65, 35, 67,55, 71). At 10 %significance level, test whether or not two populations have equal variance.

Answer 1

Solution:-

a)

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis H₀: σ²> 480

Alternative hypothesis H_A: σ² < 480

Formulate an analysis plan. For this analysis, the significance level is 0.01.

Analyze sample data. Using sample data, the degrees of freedom (DF), and the test statistic (X²).

DF = n - 1 = 10 -1

D.F = 9

We use the Chi-Square Distribution Calculator to find P(Χ² < 4.814) = 0.15

Interpret results. Since the P-value (0.15) is greater than the significance level (0.01), we have to accept the null hypothesis.

From the above test we do not have sufficient evidence in the favor of the claim that the population variance is less than 480.

b)

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis H₀: σ₁² = σ₂²

Alternative hypothesis H_A: σ₁² σ₂²

Formulate an analysis plan. For this analysis, the significance level is 0.10.

Analyze sample data. Using sample data, the degrees of freedom (DF), and the test statistic (F).

DF₁ = n₁ - 1 = 10 -1

D.F₁ = 9

DF₂ = n₂ - 1 = 6 -1

D.F₂ = 5

Test statistics:-

F = 0.3773

Since the first sample had the smaller standard deviation, this is a left-tailed test.

p value for the F distribution = 0.097

Interpret results. Since the P-value (0.097) is less than the significance level (0.10), we have to reject the null hypothesis.

From the above test there is insufficient evidence to conclude that two populations have equal variance.