Question

In 2019, 15.9% of Broadway actors were acting in their first role on Broadway. Suppose we took a survey of 38 Broadway actors and found that 18.4% of the actors we surveyed were first-timers. What are the mean and standard deviation for the sampling distribution of pÌ p^? 1. Mean: 0.159, Standard Deviation: 0.0592. Mean: 0.159, Standard Deviation: 0.36573. Mean: 0.184, Standard Deviation: 0.0634. Mean: 0.184, Standard Deviation: 0.0595. Mean: 0.159, Standard Deviation: 0.063In 2019 the CDC reported that 14.0% of US adults are smokers. Suppose you take a random sample of 30 smokers and find that the proportion of them who are current smokers is 16.7%.What is the mean and the standard deviation of the sampling distribution of pÌ p^ ?1. mean = 0.140, standard deviation = 0.0682. mean = 0.167, standard deviation = 0.0633. mean = 0.140, standard deviation = 0.0634. mean = 0.167, standard deviation = 0.068

Answer 1

For the Broadway actors acting in their first role on Broadway, mean: 0.184, Standard Deviation: 0.063.

For the proportion of smokers, mean = 0.167, standard deviation = 0.068

Explanation:

Central Limit Theorem

The Central Limit Theorem establishes 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)}`

Suppose we took a survey of 38 Broadway actors and found that 18.4% of the actors we surveyed were first-timers.

This means that
`p = 0.184, n = 38`

What are the mean and standard deviation for the sampling distribution of p^?

Mean:

`\mu = p = 0.184`

Standard deviation:

`s = \sqrt{(p(1-p))/(n)} = \sqrt{(0.184*0.816)/(38)} = 0.063`

Suppose you take a random sample of 30 smokers and find that the proportion of them who are current smokers is 16.7%.

This means that
`n = 30, p = 0.167`

What is the mean and the standard deviation of the sampling distribution of p^ ?

Mean:

`\mu = p = 0.167`

Standard deviation:

`s = \sqrt{(p(1-p))/(n)} = \sqrt{(0.167*0.833)/(30)} = 0.068`