Question

Consider independent random samples from two populations that are normal or approximately normal, or the case in which both sample sizes are at least 30. Then, if σ₁ and σ₂ are unknown but we have reason to believe that σ₁ = σ₂, we can pool the standard deviations. Using sample sizes n₁ and n₂, the sample test statistic x₁ − x₂ has a Student's t distribution where

t =

x1 − x2

s 1 n1 + 1 n2

with degrees of freedom d.f. = n₁ + n₂ − 2

and where the pooled standard deviation s is
s =

(n1 − 1)s12 + (n2 − 1)s22 n1 + n2 − 2

Note: With statistical software, select the pooled variance or equal variance options.

(a) There are many situations in which we want to compare means from populations having standard deviations that are equal. This method applies even if the standard deviations are known to be only approximately equal. Consider a report regarding average incidence of fox rabies in two regions. For region I, n₁ = 14, x₁ ≈ 4.71, and s₁ = 2.84 and for region II, n₂ = 13, x₂ ≈ 3.95, and s₂ = 2.45. The two sample standard deviations are sufficiently close that we can assume σ₁ = σ₂. Use the method of pooled standard deviation to consider the report, testing if there is a difference in population mean average incidence of rabies at the 5% level of significance. (Round your answer to three decimal places.)

t =

Find (or estimate) the P-value.

P-value > 0.5000.250 < P-value < 0.500    0.100 < P-value < 0.2500.050 < P-value < 0.1000.010 < P-value < 0.050P-value < 0.010

Conclusion

Reject the null hypothesis, there is insufficient evidence to show a difference in mean average incidence of rabies.Fail to reject the null hypothesis, there is insufficient evidence to show a difference in mean average incidence of rabies.    Reject the null hypothesis, there is sufficient evidence to show a difference in mean average incidence of rabies.Fail to reject the null hypothesis, there is sufficient evidence to show a difference in mean average incidence of rabies.

(b) Compare the t value calculated in part (a) using the pooled standard deviation with the t value calculated using the unpooled standard deviation. Compare the degrees of freedom for the sample test statistic. Compare the conclusions.

The t value for the unpooled method is much larger. The degrees of freedom are the same. The conclusions are the same.The t values are very similar. The degrees of freedom are the same. The conclusions are the same.    The t values are very similar. The degrees of freedom are the same. We reject the null using unpooled s but fail to reject using pooled s.The t values are very similar. The degrees of freedom for the unpooled method is much larger. The conclusions are the same.

Answer 1

a) pooled method

b) unpooled method

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