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

Answer:

Step-by-step 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;$\frac{p(1-p)}{n}$)

Where p' is the sample proportion

p is the population proportion and the mean of the distribution

$\frac{p(1-p)}{n}$ is the variance of the distribution

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

You can estimate it using the sample proportion:

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

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

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

I hope it helps!