## 95% conf. int. for prop. of pop. w/ wife taller than hubby

Standard deviation, mean, variance, z-scores, t-tests, etc.
maggiemagnet
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### 95% conf. int. for prop. of pop. w/ wife taller than hubby

From a large population of married couples, researchers selected a random sample of four hundred couples.

The heights of the married men are approximately normally distributed, with a mean of 70 inches and a standard deviation of 3.

The heights of the married women are approximately normally distributed, with a mean of 65 inches and a standard deviation of 2.5.

There were twenty couples in which the wife was taller than her husband; in the other 380 couples, the husband was taller than his wife.

Find a 95% confidence interval for the proportion of the population for which the wife is taller than her husband.

stapel_eliz
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I'm not good at this stuff, but let's give it a whirl.

The proportion is 20/400 = 2/40 = 1/20 = 0.05, so p = 0.05 and q = 0.95.

The z-score for 0.95 is 1.96 (or thereabouts; your book may use a different number of decimal places). So the margin of error is:

. . . . .$z\times\sqrt{\frac{pq}{n}}\, =\, (1.96)\sqrt{\frac{0.0475}{400}}\, \approx\, 0.0213586048...$

...or about 0.0214. Then the interval should be 0.05 +/- 0.0214.