SOLUTION: general motors reports that 70 percent of leased vehicles are returned with less than 36,000 miles. a recent sample of 200 vehicles returned at the end of their lease showed 158 ha

Question 1162380: general motors reports that 70 percent of leased vehicles are returned with less than 36,000 miles. a recent sample of 200 vehicles returned at the end of their lease showed 158 had less than 36,000 miles. has the proportion increased?
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
population returned within 36000 miles proportion = p = .7
population returned not within 36000 miles proportion = q = .3 = 1 - p
sample size = n = 200
s = sqrt(p * q * n) = sqrt(.7 * .3 * 200) = 6.480740698 = standard error = standard deviation of the distribution of sample means.
population mean = m = p * n = .7 * 200 = 140
sample mean = x = 158
z = (x - m) / s = (158 - 140) / 6.480740698 = 2.777460299
sample alpha = .0027393348
at 99% confidence limit the one tailed alpha would be equal to .01.
.0027... is well below that.
.0027... is even below the two tailed alpha of .005
there is a strong possibility that the proportion has increased and that the sample mean encountered is not due to random variation in the sample mean.

this can also be verified by z-score.
the critical z-score for a one tailed confidence limit of 1% would be equal to 2.3263470877.
the critical z-score for a two tailed confidence limit on the high end would be equal to 2.575829303
the z-score for the sample mean was 2.777460299.
this exceeded the critical z-score for a one tailed 99% confidence limit by a pretty good amount.
it also exceeded the critical z-score for a two tailed 99% confidence limit.
this confirms the alpha analysis, as it should.
it's a very strong indication that the proportion of vehicles that had less than 36,000 when they were turned in has actually increased and is not due to random variations in the mean of 200 vehicle samples.

the rationale is that a mean of 158 out of a sample of 200 would be expected to have a probability of occurring in only .27% of the samples taken.
the cutoff was .5% (.005) of the samples taken.
since this .27% was less than .5%, the indications were that the sample mean was due to an increase in the proportion of vehicles that had less than 36000 miles on them when they were returned, rather than due to random variations of the mean expected in a 200 vehicle sample.

not sure if i worded this correctly, but the short answer is that the proportion looks very much like it had actually increased.

RELATED QUESTIONS
A small dealership leased 21 Suburu Outbacks on 2-year leases. Two years later, when the (answered by Theo)

A small dealership leased 21 Suburu Outbacks on 2-year leases. Two years later, when the... (answered by Boreal)

I'm dying a slow death in statistics. Please help! In 2004, a small dealership leased (answered by stanbon)

According to a leasing firm's reports, the mean number of miles driven annually in its... (answered by ikleyn)

College Costs The mean undergraduate cost for tuition, fees, room, and board for... (answered by Boreal)

You work in the leasing office of GE capital and wish to predict the average mileage... (answered by Theo)

In a recent production of automobiles, 10% are found to have a minor engine defect. A... (answered by 296714)

A recent study reports that 2.5% of elementary children are allergic to peanuts. A random (answered by stanbon)