Answer:
(0.4578 , 0.5318)
Step-by-step explanation:
The attached figure shows the formula for calculating confidence intervals for the difference of proportions in large samples.
Let's call
[tex]p_1[/tex] = proportion of married couples, in the first sample, who had two or more personality preferences in common.
[tex]p_1[/tex] = 197/347 = 0.5677
[tex]q_1 = 1-p_1[/tex]
[tex]n_1[/tex] = size of the first random sample
[tex]n_1 = 347[/tex]
[tex]p_2[/tex] = proportion of married couples, in the second sample, who had no preferences in common.
[tex]p_2[/tex] = 39/535 = 0.0729
[tex]q_2 = 1-p_2[/tex]
[tex]n_2[/tex] = size of the second random sample = 535
[tex]100 (1-\alpha)[/tex] = confidence%.
[tex]100 (1-\alpha)[/tex] = 80%
[tex](1-\alpha) = 0.8[/tex]
[tex]\alpha = 0.2[/tex]
Looking in the normal standard table, we have that [tex]Z_ {0.2 / 2}[/tex] = 1.28.
Substituting this values in the formula we have:
[tex]0.5677-0.0729 + 1.28\sqrt{\frac{0.5677(1-0.5677)}{347}+\frac{0.0729(1-0.0729)}{535}}= 0.4948 + 0.03696\\\\ 0.5677-0.0729 - 1.28\sqrt{\frac{0.5677(1-0.5677)}{347}+\frac{0.0729(1-0.0729)}{535}}= 0.4948 - 0.03696[/tex]
[tex]0.4948 + 0.03696 = 0.5318\\\\0.4948 - 0.03696 = 0.4578\\[/tex]
Then the confidence interval for p1-p2 is: (0.4578 , 0.5318)
