Question

How many commuters must be randomly selected to estimate the mean driving time of Chicago commuters? We want 90% confidence that the sample mean is within 4 minutes of the population mean, and the population standard deviation is known to be 12 minutes.

Answer 1

The number of commuters to be randomly selected to estimate the mean driving time of Chicago commuters is 25

Explanation:

At 90% confidence

we have

Minimum sample size, n given by

`n = \left ((z_(c)\sigma )/(E) \right )^(2)`

Where:

c = Confidence level = 90%

σ = Standard deviation = 12 minutes

E = Maximum error in estimate = minutes

-
`z_c` = 1.64 and
`z_c` = 1.64

Therefore,
`z_c` = 0.51994 and therefore, n is given by

`n = \left ((1.64* 12 )/(4) \right )^(2)` = 24.2064

As we are talking of population, we round to the next whole number, thus;

n = 25 commuters.

Answer 2

`n=((1.640(12))/(4))^2 =24.206 \approx 25`

So the answer for this case would be n=25 rounded up to the nearest integer

Explanation:

Previous concepts

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".

The margin of error is the range of values below and above the sample statistic in a confidence interval.

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

Solution to the problem

The margin of error is given by this formula:

`ME=z_(\alpha/2)(\sigma)/(√(n))` (a)

And on this case we have that ME =4 and we are interested in order to find the value of n, if we solve n from equation (a) we got:

`n=((z_(\alpha/2) \sigma)/(ME))^2` (b)

The critical value for 90% of confidence interval now can be founded using the normal distribution. And in excel we can use this formla to find it:"=-NORM.INV(0.05;0;1)", and we got
`z_(\alpha/2)=1.640`, replacing into formula (b) we got:

`n=((1.640(12))/(4))^2 =24.206 \approx 25`

So the answer for this case would be n=25 rounded up to the nearest integer