Question

Coping with No Success: According to the Rule of Three, when we have a sample size n with x = 0 successes, we have 95% confidence that the true population proportion has an upper bound of 3/n. (See “A Look at the Rule of Three,” by Jovanovic and Levy, American Statistician, Vol. 51, No. 2.)a. If n independent trials result in no successes, why can’t we find confidence interval limits by using the methods described in this section? b. If 40 couples use a method of gender selection and each couple has a baby girl, what is the 95% upper bound for p, the proportion of all babies who are boys?

Short answer

a.If there are no successes in n independent trials, then the requirement of at least 5 successes in the sample is not satisfied. Hence the methods used in this section cannot be used to construct a confidence interval when the number of successes is equal to 0.

b.The 95% upper bound of the proportion of boys is equal to 0.075.

Step-by-step solution

Given information

It is given that the Rule of Three is utilized for finding the upper limit of the confidence interval when the number of successes is equal to 0.

Also, in a sample of 40 couples, each couple has a baby girl. The 95% upper bound for the population proportion (p) of boys is to be computed.

Requirement for computing the confidence interval

a.

One of the requirements for constructing a confidence interval estimate of the population proportion (using the formula given in the chapter) is that there should be at least 5 successes and at least 5 failures in that sample.

For a sample that has 0 successes and a confidence interval is to be constructed for the proportion of successes, the method discussed in the chapter cannot be used as the above-stated requirement is not fulfilled.

Therefore, the given formula cannot be applied.

95% upper bound for the population proportion

b.

Let success be defined as having a baby boy.

Let p denote the population proportion of baby boys.

Here, the sample considered has no couple with a baby boy.

Thus, the number of successes is equal to 0.

Using the Rule of Three, which states that for a sample with n independent trials and number of successes equal to 0, the 95% upper bound of the population proportion has the following value:

$\frac{3}{n}$

For the given question, the value of n is equal to 40.

Using this rule, the 95% upper bound of p is computed below:

$$\begin{gathered} \frac{3}{n}=\frac{3}{40} \\ =0.075 \end{gathered}$$

Therefore, the 95% upper bound for the population proportion of boys is equal to 0.075.