Question

In the United States, 7% of all registered voters belong to the Green party. A random sample of 50 registered voters is taken. Use this information to answer the following four questions. What is the expected value of the sample proportion?

1. Determine P(p < .02).
2. Determine P(p > .15).
3. Determine P(.05 < p < .09).

Answer 1

The expected value of the sample proportion is of 0.07.

1. P(p < .02) = 0.0823

2. P(p > .15) = 0.0132

3. P(.05 < p < .09) = 0.4176

Explanation:

This question is solved using the normal probability distribution and the central limit theorem.

Normal probability distribution

When the distribution is normal, we use the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

Central Limit Theorem

For a proportion p in a sample of size n, the sampling distribution of the sample proportion will be approximately normal with mean
`\mu = p` and standard deviation
`s = \sqrt{(p(1-p))/(n)}`

7% of all registered voters belong to the Green party. 50 voters:

This means that
`p = 0.07, n = 50`

So, for the normal distribution:

`\mu = 0.07, s = \sqrt{(p(1-p))/(n)} = \sqrt{(0.07*0.93)/(50)} = 0.036`

The expected value of the sample proportion is of 0.07.

1. Determine P(p < .02).

This is the pvalue of Z when X = 0.02. So

`Z = (X - \mu)/(\sigma)`

By the Central Limit Theorem

`Z = (X - \mu)/(s)`

`Z = (0.02 - 0.07)/(0.036)`

`Z = -1.39`

`Z = -1.39` has a pvalue of 0.0823

So

P(p < .02) = 0.0823

2. Determine P(p > .15).

This is 1 subtracted by the pvalue of Z when X = 0.15. So

`Z = (X - \mu)/(s)`

`Z = (0.15 - 0.07)/(0.036)`

`Z = 2.22`

`Z = 2.22` has a pvalue of 0.9868

1 - 0.9868 = 0.0132

So

P(p > .15) = 0.0132

3. Determine P(.05 < p < .09).

This is the pvalue of Z when X = 0.09 subtracted by the pvalue of Z when X = 0.05. So

X = 0.09

`Z = (X - \mu)/(s)`

`Z = (0.09 - 0.07)/(0.036)`

`Z = 0.55`

`Z = 0.55` has a pvalue of 0.7088

X = 0.05

`Z = (X - \mu)/(s)`

`Z = (0.05 - 0.07)/(0.036)`

`Z = -0.55`

`Z = -0.55` has a pvalue of 0.2912

0.7088 - 0.2912 = 0.4176. So

P(.05 < p < .09) = 0.4176