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 700 customers indicates that 76% 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 175 customers (assuming the proportion is about the same)?

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JeanaShupp JeanaShupp · Answer 1 · 2019-10-08T12:58:16+02:00

Answer: a) 0.0161

b) 0.0323

Step-by-step explanation:

The standard deviation of the sampling distribution of the proportion :

[tex]\sigma_p=\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}[/tex]

a ) Given : n=700 and [tex]\hat{p}=0.76[/tex]

Then, the standard deviation of the sampling distribution of the proportion:

[tex]\sigma_p=\sqrt{\dfrac{0.76(1-0.76)}{700}}=0.0161422250192\approx0.0161[/tex]

Hence, the estimated standard deviation of the sampling distribution of the proportion =0.0161

b) If n= 175 and [tex]\hat{p}=0.76[/tex]

Then, the standard deviation of the sampling distribution of the proportion:

[tex]\sigma_p=\sqrt{\dfrac{0.76(1-0.76)}{175}}=0.0322844500385\approx0.0323[/tex]

Hence, the estimated standard deviation have been if he had surveyed only 175 customers= 0.0323