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Chapter 13: Problem 21

A random sample of the ages of 14 brides and their grooms showed that in 10 of the pairs the grooms were older, in 1 pair they were the same age, and in 3 pairs the bride was older. Perform a sign test with a significance level of \(0.05\) to test the hypothesis that grooms tend to be older than their brides.

Short Answer

Expert verified
The sign test results do not provide strong enough evidence to reject the null, therefore we can't say with a significance level of 0.05 that grooms tend to be older than brides.

Step by step solution

01

Define the hypothesis

The null hypothesis (H0) is: grooms and brides are of the same age on average. The alternative hypothesis (Ha) is: grooms tend to be older than brides.
02

Obtain the test statistic from the sample

The test statistic for a sign test is the smaller of the numbers of negative and positive differences (signs). Here, there are 10 cases where the groom is older (positive), 3 where the bride is older (negative) and 1 pair is of the same age. The same ages are generally excluded from a sign test. Hence, the test statistic is min(10,3) = 3.
03

Determine the critical region

We need to calculate the critical region, which is the region that, if the test statistic falls in it, we reject the null hypothesis. We use a binomial distribution B(n,p) with n= total no. of pairs, n=13 and p= probability of success (groom is older), p=0.5. For a level 0.05 two-sided test, the rejection regions would be the smallest and largest 2.5 percent of the distribution. Given our n=13, we need to calculate the cumulative binomial probability until we reach or exceed 0.025 in both tails to find the critical region.
04

Decision

The test statistic (3) does not fall into the critical region and hence we fail to reject the null hypothesis. This means that there is not strong enough statistical evidence at the 0.05 significance level to conclude that grooms tend to be older than their brides.

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