Question

15. Test the hypothesis that at the at 0.05 level of significance for the given sample data Assume that the populations are normally distributed Chapter 111 Population 1 Population 2 11 11 n 5 2.5 a. Write both alternative and null hypothesis. H: 01-02 Ho: 01-02 Two tail test b. Find test statistics (show formula and final answer with two decimals) F = 5,7/S7 = 2.52/4.62 = 0.30 c. Find P-value (round to 3 decimal places) 0.965 > 0.05. We do not reject Ho d. Write the conclusion in the contest of the problem At a 0.05, there is insufficient evidence to prove that 01 02

Answer 1

Let s₁ and s₂ denote the sample standard deviations and n₁ and n₂ denote the sample sizes of population1 and population 2 respectively.

s₁ = 2.5 s₂ = 4.6 n₁ = n₂ = 11

a.

Null hypothesis: $H_0:\sigma_1=\sigma_2$

Alternative hypothesis: $H_1:\sigma_1\neq \sigma_2$

b.

Test statistic: $F=\frac{s_1^2}{s_2^2}=\frac{2.5^2}{4.6^2}=0.30$

# we always divide larger variance by smaller variance.

As the populations are normally distributed under H₀,test statistic follows F distribution with (n₁-1, n₂-1) = (10,10) degrees of freedom.

c.

P-value = P[ f_10,10 > 0.30 ] = 0.96455 = 0.965

You can use following R command to find the p-value

#1-pf(0.3,10,10)

d.

We reject H₀ at 0.05 level of significance if p-value < 0.05

Here, P-value > 0.05

Hence, we do not reject H₀.

We do not have enough evidence to reject H₀. That is there is insufficient evidence to prove that $\sigma_1\neq \sigma_2$

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