Question

Consider the following sample data drawn independently from normally distributed populations with equal population variances.

Sample 1	Sample 2
11.2	11.4
11.5	12.1
7.7	12.7
10.7	10.2
10.2	10.2
9.1	9.9
9.3	10.9
11.6	12.7

a. Construct the relevant hypotheses to test if the mean of the second population is greater than the mean of the first population.

• a) H₀: μ₁ − μ₂ = 0; H_A: μ₁ − μ₂ ≠ 0

• b) H₀: μ₁ − μ₂ ≥ 0; H_A: μ₁ − μ₂ < 0

• c) H₀: μ₁ − μ₂ ≤ 0; H_A: μ₁ − μ₂ > 0

b-1. Calculate the value of the test statistic. (Negative values should be indicated by a minus sign. Round all intermediate calculations to at least 2 decimal places and final answer to 2 decimal places.)

Test statistic =   

b-2. Calculate the critical value at the 5% level of significance. (Negative value should be indicated by a minus sign. Round your answer to 3 decimal places.)

Critical value =   

b-3. Using the critical value approach, can we reject the null hypothesis at the 5% significance level?

• a) Yes, since the value of the test statistic is less than the critical value of -1.761.

• b) Yes, since the value of the test statistic is less than the critical value of -2.145.

• c) No, since the value of the test statistic is not less than the critical value of -2.145.

• d) No, since the value of the test statistic is not less than the critical value of -1.761.

c. Using the critical value approach, can we reject the null hypothesis at the 10% level?

• a) Yes, since the value of the test statistic is less than the critical value of -1.345.

• b) Yes, since the value of the test statistic is less than the critical value of -1.761.

• c) No, since the value of the test statistic is not less than the critical value of -1.345.

• d) No, since the value of the test statistic is not less than the critical value of -1.761.

Answer 1

a)

b) H₀: μ₁ − μ₂ ≥ 0; H_A: μ₁ − μ₂ < 0

b-1)

	sample 1		sample 2
x1    =	10.16	x2      =	11.26
s1    =	1.37	s2      =	1.14
n1    =	8	n2     =	8
Pooled Variance Sp2=((n1-1)s21+(n2-1)*s22)/(n1+n2-2)=				1.5898

Point estimate : x1-x2=			-1.1000
standard error se =Sp*√(1/n1+1/n2)=			0.630
test stat t =(x1-x2-Δo)/Se=		-1.74

b-2)

for 0.05 level ,left tail test &14 df, critical t=

-1.761

b3)

d) No, since the value of the test statistic is not less than the critical value of -1.761.

c)

a) Yes, since the value of the test statistic is less than the critical value of -1.345.