A market researcher for a provider of iPod accessories wants to know the proport

Question

A market researcher for a provider of iPod accessories wants to know the proportion of customers who own cars to assess the market for a new iPod car charger. A survey of 500 customers indicates that 76% own cars. Find the sampling distribution of p^hat. What is the expected number of customers who own cars? Find the probability that less than 75% of customers own cars. Find the middle 90% of the proportion of these 500 customers who own cars. What is the probability that more than 400 customers own cars?

Explanation / Answer

a) sampling distribution of p is normal with point estimate p=0.76

b) expected number =np=500*0.76=380

c)here std error of proportion =(p(1-p)/n)1/2 =0.0191

hence P(X<.75)=P(Z<(0.75-0.76)/0.0191)=P(Z<-0.5236)=0.3003

d) for middle 90% ; lies at 5% and 95% for which critical value of z=-/+1.6449

hence confidence interval =sample proportion -/+ z*std error =0.7286 ; 0.7914

e) for mean number =np=500*76=380

std deviation =(np(1-p))1/2 =9.55

hence P(X>400) =1-P(X<=400)=1-P(Z<(400.5-380)/9.55)=1-P(Z<2.1466)=1-0.9841=0.0159

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