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 thesame)?

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cchilabert cchilabert · Answer 1 · 2020-03-16T03:18:43+01:00

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;[tex]\frac{p(1-p)}{n}[/tex])

Where p' is the sample proportion

p is the population proportion and the mean of the distribution

[tex]\frac{p(1-p)}{n}[/tex] is the variance of the distribution

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

You can estimate it using the sample proportion:

[tex]\sqrt{\frac{p'(1-p')}{n}} = \sqrt{\frac{0.73*0.27}{600} }=0.0181[/tex]

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

[tex]\sqrt{\frac{p'(1-p')}{n}} = \sqrt{\frac{0.73*0.27}{150} }=0.0362[/tex]

I hope it helps!