Answer:
critical value = 1.645
The 90% confidence interval = ( -22.62, -17.58)
Step-by-step explanation:
Given that:
the sample size [tex]n_1[/tex] = 178
the sample size [tex]n_2[/tex] = 226
the sample mean [tex]\overline x_1[/tex] = 54.4
the sample mean [tex]\overline x_2[/tex] = 74.5
population standard deviation [tex]\sigma_1[/tex] = 18.58
population standard deviation [tex]\sigma_2[/tex] = 9.52
level of significance ∝ = 1 - 0.90 = 0.10
The critical value for [tex]Z_{\alpha/2} = Z _{0.10/2} = Z_{0.005}[/tex] is 1.645
For the construction of our confidence interval, we use 90% since that is used to find the critical value.
∴
The margin of error = [tex]Z \times\sqrt{\dfrac{\sigma_1^2}{n_1} + \dfrac{\sigma_2^2}{n_2}}[/tex]
[tex]1.645 \times\sqrt{\dfrac{18.58^2}{178} + \dfrac{9.52^2}{226}}[/tex]
[tex]1.645 \times\sqrt{\dfrac{345.2164}{178} + \dfrac{90.6304}{226}}[/tex]
[tex]1.645 \times\sqrt{2.34042}[/tex]
[tex]\simeq[/tex] 2.52
The lower limit = [tex]( \overline x_1 - \overline x_2) - (M.O.E)[/tex]
= ( 54.4-74.5) - (2.52)
= -20.1 - 2.52
= -22.62
The upper limit = [tex]( \overline x_1 - \overline x_2) + (M.O.E)[/tex]
= ( 54.4-74.5) + (2.52)
= -20.1 + 2.52
= -17.58
The 90% confidence interval = ( -22.62, -17.58)
