A sample of 1,400 American households was asked if they planned to buy a new car next year.Of the respondents,34% indicated they planned to buy a new car next year.Construct a 90% confidence interval of the proportion of American households who expect to buy a new car next year.
A) [0.3152,0.3648]
B) [0.3191,0.3609]
C) [0.3394,0.3406]
D) [0.3354,0.3446]

Question

Quizplus · Accepted Answer

To construct a confidence interval for a proportion, we use the formula:

point estimate ± margin of error

where the point estimate is the sample proportion (p̂) and the margin of error is calculated using the formula:

z* * √((p̂(1-p̂))/n)

where z* is the z-score from the standard normal distribution corresponding to the desired level of confidence (90% in this case), n is the sample size, and p̂ is the sample proportion.

Plugging in the given values:

- point estimate = p̂ = 0.34 
- z* for a 90% confidence interval is approximately 1.645 
- n = 1400

We get:

0.34 ± 1.645 * √((0.34(1-0.34))/1400)

Simplifying, we get:

0.34 ± 0.021

So the 90% confidence interval for the proportion of American households who plan to buy a new car next year is:

[0.319,0.360]

The closest choice is B.

A Sample of 1,400 American Households Was Asked If They