Question

Suppose 41%41% of American singers are Grammy award winners. If a random sample of size 860860 is selected, what is the probability that the proportion of Grammy award winners will differ from the singers proportion by less than 5%5%?

Answer 1

Answer:

99.72% probability that the proportion of Grammy award winners will differ from the singers proportion by less than 5%.

Step-by-step explanation:

To solve this question, we need to understand 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:

$Z = \frac{X - \mu}{\sigma}$

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 $s = \frac{\sigma}{\sqrt{n}}$.

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 $s = \sqrt{\frac{p(1-p)}{n}}$

In this question:

$n = 860, p = 0.41$

So

$\mu = 0.41, s = \sqrt{\frac{0.41*0.59}{860}} = 0.0168$

What is the probability that the proportion of Grammy award winners will differ from the singers proportion by less than 5%?

This is the pvalue of Z when X = 0.41 + 0.05 = 0.46 subtracted by the pvalue of Z when X = 0.41 - 0.05 = 0.36. So

X = 0.46

$Z = \frac{X - \mu}{\sigma}$

By the Central Limit Theorem

$Z = \frac{X - \mu}{s}$

$Z = \frac{0.46 - 0.41}{0.0168}$

$Z = 2.98$

$Z = 2.98$ has a pvalue of 0.9986

X = 0.36

$Z = \frac{X - \mu}{s}$

$Z = \frac{0.36 - 0.41}{0.0168}$

$Z = -2.98$

$Z = -2.98$ has a pvalue of 0.0014

0.9986 - 0.0014 = 0.9972

99.72% probability that the proportion of Grammy award winners will differ from the singers proportion by less than 5%.