Question

A random sample of 366 married couples found that 286 had two or more personality preferences in common. In another random sample of 552 married couples, it was found that only 38 had no preferences in common. Let p₁ be the population proportion of all married couples who have two or more personality preferences in common. Let p₂ be the population proportion of all married couples who have no personality preferences in common.

(a) Find a 90% confidence interval for p₁ – p₂. (Use 3 decimal places.)

lower limit
upper limit

(b) Explain the meaning of the confidence interval in part (a) in the context of this problem. Does the confidence interval contain all positive, all negative, or both positive and negative numbers? What does this tell you (at the 90% confidence level) about the proportion of married couples with two or more personality preferences in common compared with the proportion of married couples sharing no personality preferences in common?

Because the interval contains both positive and negative numbers, we can not say that a higher proportion of married couples have two or more personality preferences in common.

We can not make any conclusions using this confidence interval.    

Because the interval contains only negative numbers, we can say that a higher proportion of married couples have no personality preferences in common.

Because the interval contains only positive numbers, we can say that a higher proportion of married couples have two or more personality preferences in common.

Answer 1

(a) = 286/ 366 = 0.7814, 1 - = 0.2186, n₁ = 366,

= 38 / 552 = 0.0688, 1 - = 0.9312, n₂ = 552,

The Zcritical (2 tail) for = 0.10, is 1.645

The Confidence Interval is given by (- ) ME, where

(- ) = 0.7814 – 0.0688 = 0.7126

The Lower Limit = 0.7126 - 0.0397 = 0.6729 0.673

The Upper Limit = 0.7126 + 0.0397 = 0.7523    0.752

The 90% Confidence interval is 0.673 < p1 - p2 < 0.752

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(b) The Interval contains both positive numbers.

In terms of a Hypothesis test for difference in proportions, since the upper limit and the lower limit have positive values, it means that the results are statistically significant, i.e we would reject the null hypothesis H0: p1- p2 = 0, in favour of the 2 tailed alternative Ha: p1 - p2 0.

Therefore Option 4: Because the interval contains only positive numbers, we can say that a higher proportion married couples have two or more personality preferences in common.