Question

An automotive manufacturer wants to know the proportion of new car buyers who prefer foreign cars over domestic. Suppose a sample of 390 new car buyers is drawn. Of those sampled, 105 preferred foreign over domestic cars. Using the data, construct the 80% confidence interval for the population proportion of new car buyers who prefer foreign cars over domestic cars. Round your answers to three decimal places.

Answer 1

80% confidence interval for the population proportion of new car buyers who prefer foreign cars over domestic cars is [0.240 , 0.298].

Explanation:

We are given that an automotive manufacturer wants to know the proportion of new car buyers who prefer foreign cars over domestic.

Suppose a sample of 390 new car buyers is drawn. Of those sampled, 105 preferred foreign over domestic cars.

Firstly, the pivotal quantity for 80% confidence interval for the population proportion is given by;

P.Q. =
`\frac{\hat p -p}{\sqrt{(\aht p(1-\hat p))/(n) } }` ~ N(0,1)

where,
`\hat p` = sample proportion of car buyers who preferred foreign over domestic cars =
`(105)/(390)` = 0.27

n = sample of new car buyers = 390

p = population proportion

Here for constructing 80% confidence interval we have used One-sample z proportion statistics.

So, 80% confidence interval for the population proportion, p is ;

P(-1.2816 < N(0,1) < 1.2816) = 0.80 {As the critical value of z at 10% level

of significance are -1.2816 & 1.2816}

P(-1.2816 <
`\frac{\hat p -p}{\sqrt{(\aht p(1-\hat p))/(n) } }` < 1.2816) = 0.80

P(
`-1.2816 * {\sqrt{(\aht p(1-\hat p))/(n) } }` <
`{\hat p -p}` <
`1.2816 * {\sqrt{(\aht p(1-\hat p))/(n) } }` ) = 0.80

P(
`\hat p-1.2816 * {\sqrt{(\aht p(1-\hat p))/(n) } }` < p <
`\hat p+1.2816 * {\sqrt{(\aht p(1-\hat p))/(n) } }` ) = 0.80

80% confidence interval for p = [
`\hat p-1.2816 * {\sqrt{(\hat p(1-\hat p))/(n) } }` ,
`\hat p+1.2816 * {\sqrt{(\hat p(1-\hat p))/(n) } }`]

= [
`(105)/(390) -1.2816 * {\sqrt{((105)/(390)(1-(105)/(390)))/(390) } }` ,
`(105)/(390) +1.2816 * {\sqrt{((105)/(390)(1-(105)/(390)))/(390) } }` ]

= [0.240 , 0.298]

Therefore, 80% confidence interval for the population proportion of new car buyers who prefer foreign cars over domestic cars is [0.240 , 0.298].