Question

A market researcher for a provider of music player accessories wants to know the proportion of customers who own cars to assess the market for a new car charger. A survey of 600 customers indicates that 73% own cars. ​a) What is the estimated standard deviation of the sampling distribution of the​ proportion? ​b) How large would the estimated standard deviation have been if he had surveyed only 150 customers​ (assuming the proportion is about the​same)?

Answer 1

Explanation:

Hello!

To evaluate the market for a new car charger, a market researcher surveyed 600 customers and found out that 73% of them own a car.

The parameter of interest is the population proportion and the study variable is "the number of customers that own a car out of 600".

Using the Central Limit Theorem you can approximate the distribution of the sampling proportion to normal:

p'≈N(p;
`(p(1-p))/(n)`)

Where p' is the sample proportion

p is the population proportion and the mean of the distribution

`(p(1-p))/(n)` is the variance of the distribution

Then de standard deviation of the samplig proportion distribution is
`\sqrt{(p(1-p))/(n)}`

You can estimate it using the sample proportion:

`\sqrt{(p'(1-p'))/(n)} = \sqrt{(0.73*0.27)/(600) }=0.0181`

Using the same sample proportion but a sample size of n=150, the estimated standard deviation is:

`\sqrt{(p'(1-p'))/(n)} = \sqrt{(0.73*0.27)/(150) }=0.0362`

I hope it helps!