Rachael L. answered • 02/18/22
I have been teaching AP Statistics for 17 years, and graded AP exam.
1a. since this is a percent, we know that it is a proportion. the sample n=200 and the p-hat =.25.
We now have to go through the conditions before we attempt the confidence interval. There are 3
conditions.
1) has to be random. Since it does not state that in the problem, we can assume that the
sample represents the population,
2) 10% rule for independence. a sample of 200 people is < 10% of all
people that test drive cars at the dealership.
3) n(p hat)≥ 10 and n(1-p hat)≥ 10. so, 200(.25)=50 and
200(.75)=150, so this condition is met.
Since all three are met we can now continue with the confidence interval.
The equation for a confidence interval is: p-hat ± z*(√((p-hat)(1-p-hat)/n). have to look up the z critical(z*) on chart or in your calculator which is 1.96 for 95% confidence
.25 ± 1.96(√(.25)(.75)/200)
.25 ± .06. (note: this gives us a margin of error of 6%)
(.19, .31)
Interpret: I am 95% confident that the true proportion (percentage) of test-drives that end up in a sale within one month is between 19% and 31%.
b. To find the desired sample size for a specific margin of error, you just need to use the margin of error formula from the confidence interval formula from above and solve for n.
The margin of error is the part of the equation that is after the ±. Margin of error = z*(√((p-hat)(1-p-hat)/n)
.04= 1.96(√(.25)(.75)/n)
once you solve for n algebraically you get: n = (1.96/.04)2(.25)(.75) = 450.1875 ≈ 451(you always round up)
Since 451-200 is 251, you need 251 more people in your sample to reduce the margin of error from 6% to 4%
