Question

The administration at Pierce College conducted a survey to determine the proportion of students who ride a bike to campus. Of the 125 students surveyed 6 ride a bike to campus. Which of the following is a reason the administration should not calculate a confidence interval to estimate the proportion of all students who ride a bike to campus? Check all that apply.

A. The sample needs to be random but we don’t know if it is.

B. The actual count of bike riders is too small.

C. The actual count of those who do not ride a bike to campus is too small.

D. n(p-hat) is not greater than 10.

E. n(1 minus p-hat) is not greater than 10.

Answer 1

The correct option is (D).

Explanation:

To construct the (1 - α)% confidence interval for population proportion the distribution of proportions must be approximated by the normal distribution.

A Normal approximation to binomial can be applied to approximate the distribution of proportion p, if the following conditions are satisfied:

• 
  `n\hat p \geq 10`

• 
  `n ( 1 - \hat p) \geq 10`

In this case p is defined as the proportions of students who ride a bike to campus.

A sample of n = 125 students are selected. Of these 125 students X = 6 ride a bike to campus.

Compute the sample proportion as follows:

`\hat p=(X)/(n)=(6)/(125)=0.048`

Check whether the conditions of Normal approximation are satisfied:

`n\hat p =125* 0.048=6<10\\n(1-\hat p) =125* (1-0.048)=119>10`

Since
`n\hat p <10`, the Normal approximation to Binomial cannot be applied.

Thus, the confidence interval cannot be used to estimate the proportion of all students who ride a bike to campus.

Thus, the correct option is (D).