Question

A sample survey is designed to estimate the proportion of sports utility vehicles being driven in the state of California. A random sample of 500 registrations are selected from a Department of Motor Vehicles database, and 64 are classified as sports utility vehicles. (a) Use a 95% confidence interval to estimate the proportion of sports utility vehicles in California. (Round your answers to three decimal places.)

Answer 1

Answer:
`(0.099,\ 0.157)`

Explanation:

We know that the confidence interval for population standard deviation is given by :-

`\hat{p}\pm z^*\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}`

, where , n= sample size.

`\hat{p}` = sample proprotion.

z* = critical z-value.

Given : A random sample of 500 registrations are selected from a Department of Motor Vehicles database, and 64 are classified as sports utility vehicles.

i.e. n= 500

`\hat{p}=(64)/(500)=0.128`

We know that critical z-value for 95% confidence = z*=1.96

Then, the 95% confidence interval to estimate the proportion of sports utility vehicles in California will be :-

`0.128\pm (1.95)\sqrt{(0.128(1-0.128))/(500)}`

`0.128\pm (1.95)√(0.000223232)`

`0.128\pm (1.95)(0.0149409504383)`

`0.128\pm 0.0291348533547\approx0.128\pm0.029=(0.128-0.029,\ 0.128+0.029)=(0.099,\ 0.157)`

Hence, the 95% confidence interval to estimate the proportion of sports utility vehicles in California. =
`(0.099,\ 0.157)`