We intend to estimate the average driving time of a group of commuters. From a previous study, we believe that the average time is 42 minutes with a standard deviation of 10 min…

Question

asked Oct 12, 2022 220k views

1 Answer

Kelvin Hu · Answer 1 · 2022-10-17T14:13:35+0000

Answer:

The smallest sample size that we should consider is 68.

Explanation:

We have that to find our
$\alpha$ level, that is the subtraction of 1 by the confidence interval divided by 2. So:

$\alpha = (1 - 0.9)/(2) = 0.05$

Now, we have to find z in the Ztable as such z has a pvalue of
$1 - \alpha$ .

That is z with a pvalue of
1 - 0.05 = 0.95 , so Z = 1.645.

Now, find the margin of error M as such

$M = z(\sigma)/(√(n))$

In which
$\sigma$ is the standard deviation of the population and n is the size of the sample.

Standard deviation of 10 minutes.

This means that
$\mu = 10$

What is the smallest sample size that we should consider?

This is n for which M = 2. So

$M = z(\sigma)/(√(n))$

2 = 1.645(10)/(√(n))

2√(n) = 1.645*10

√(n) = 1.645*5

(√(n))^2 = (1.645*5)^2

n = 67.7

Rounding up:

The smallest sample size that we should consider is 68.