Question

Use the Central Limit Theorem to find the indicated probability. The sample size is n, the population proportion is p, and the sample proportion is p hat . Round to four decimal places and please explain how you got this answer. n = 160, p = 0.29; P( phat > 0.3)

Answer 1

Applying the Central Limit Theorem for proportions, P( phat > 0.3) = 0.3897

Explanation:

Normal probability distribution

Problems of normally distributed samples are solved using 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

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean
`\mu` and standard deviation
`\sigma`, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
`\mu` and standard deviation
`s = (\sigma)/(√(n))`.

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

For a proportion p in a sample of size n, we have that
`\mu = p, s = \sqrt{(p(1-p))/(n)}`

In this problem, we have that:

`p = 0.29, \mu = 0.29, n = 160, s = \sqrt{(0.29*0.71)/(160)} = 0.0359`

P( phat > 0.3)

This is 1 subtracted by the pvalue of Z when X = 0.3. So

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

By the Central Limit Theorem

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

`Z = (0.3 - 0.29)/(0.0359)`

`Z = 0.28`

`Z = 0.28` has a pvalue of 0.6103

1 - 0.6103 = 0.3897

Applying the Central Limit Theorem for proportions, P( phat > 0.3) = 0.3897