Question

According to a 2009 Reader's Digest article, people throw away approximately 14% of what they buy at the grocery store. Assume this is the true proportion and you plan to randomly survey 154 grocery shoppers to investigate their behavior. What is the probability that the sample proportion exceeds 0.08

Answer 1

98.40% probability that the sample proportion exceeds 0.08

Explanation:

To solve this question, we need to understand the normal probability distribution and the central limit theorem.

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

In this question, we have that:

`n = 154, \mu = 0.14, s = \sqrt{(0.14*0.86)/(154)} = 0.02796`

What is the probability that the sample proportion exceeds 0.08

This is 1 subtracted by the pvalue of Z when X = 0.08. So

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

By the Central Limit Theorem

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

`Z = (0.08 - 0.14)/(0.02796)`

`Z = -2.145`

`Z = -2.145` has a pvalue of 0.0160

1 - 0.0160 = 0.9840

98.40% probability that the sample proportion exceeds 0.08