Answer:
The 90% confidence interval for (u1 - u2) is the range from; ( 0.50 - 2.06 )
Step-by-step explanation:
Given the data in the question;
Adult with No children under 18 Adult with children under 18
x"₁ = 5.65 x"₂ = 4.37
s₁ = 2.43 s₂ = 1.73
n₁ = 40 n₂ = 40
with 90% confidence
significance level ∝ = 1 - 90% = 1 - 0.9 = 0.1
Degree of freedom df = 39
Now, since variance are not equal and are also unknown, we will use 2 sample t-distribution method;
Standard Error: σx"₁ - x"₂ = [tex]\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2} }[/tex]
Critical value = [tex]t_{\alpha /2, df[/tex] = [tex]t_{0.05, df=39[/tex] = 1.645 { from standard normal table }
Now,
Margin of Error = E = [tex]t_{\alpha /2, df[/tex] × [tex]\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2} }[/tex]
we substitute
E = 1.645 × [tex]\sqrt{\frac{2.43^2}{40} + \frac{1.73^2}{40} }[/tex]
E = 1.645 × √( 0.1476225 + 0.0748225 )
E = 0.7758478
Now, our point difference will be; x"₁ - x"₂ = 5.65 - 4.37 = 1.28
So Limit at 90% confidence interval will be;
x"₁ - x"₂ ± E
Lower Limit = x"₁ - x"₂ - E = 1.28 - 0.7758478 = 0.50415 ≈ 0.50
Upper Limit = x"₁ - x"₂ + E = 1.28 + 0.7758478 = 2.05584 ≈ 2.06
Therefore, The 90% confidence interval for (u1 - u2) is the range from; ( 0.50 - 2.06 )
