Question

In order to compare the means of populations, independent random samples of 390 observations are selected from each population, with the results found in the table below

x1=5283, x2=5242, s1=143, s2=194

Use a​ 95% confidence interval to estimate the difference between the population means (u1-u2)

a) find the confidence interval

b) Test the null hypothesis Ho: (u1-u2)=0 vs the alternative hypothesis Ha: (u1-u2)=/=0. Use a=0.05. What is the test statistic? What is the p-value?

c) Test the null hypothesis Ho: (u1-u2)>0 What is the p-value?

d) Test the null hypothesis Ho: (u1-u2)=30 vs the alternative hypothesis Ha: (u1-u2)=/=30. Use a=0.05. What is the test statistic? What is the p-value?

Answer 1

a) confidence interval is

μ₁ - μ₂ = (M₁ - M₂) ± ts_{(M1 - M2)}

where:

M₁ & M₂ = sample means
t = t statistic determined by confidence level
s_{(M1 - M2)} = standard error = √((s²_p/n₁) + (s²_p/n₂))

Calculation

Pooled Variance
s²_p = ((df₁)(s²₁) + (df₂)(s²₂)) / (df₁ + df₂) = 22595065 / 778 = 29042.5

Standard Error
s_{(M1 - M2)} = √((s²_p/n₁) + (s²_p/n₂)) = √((29042.5/390) + (29042.5/390)) = 12.2

Confidence Interval
μ₁ - μ₂ = (M₁ - M₂) ± ts_{(M1 - M2)} = 41 ± (1.96 * 12.2) = 41 ± 23.96

, 95% CI [17.04, 64.96]

b)

p-value=0.0008

c)

p-value =0.0004 [right tail,one tail test

d)

p-value=0.3677