Question

The state highway department is studying traffic patterns on one of the busiest highways in the state. As part of the study, the department needs to estimate the average number of vehicles that pass an intersection each day. A random sample of 64 days gives us a sample mean of 14,205 cars and a sample standard deviation of 1,010 cars. After calculating the confidence interval, the highway department officials decide that the precision is too low for their needs. They feel the precision should be 300 cars. Given this precision, and needing to be 99 percent confident, how many days do they need to sample

Answer 1

Answer:

The need to sample 76 days.

Step-by-step explanation:

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

$\alpha = \frac{1-0.99}{2} = 0.005$

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

So it is z with a pvalue of $1-0.005 = 0.995$, so $z = 2.575$

Now, find the margin of error M as such

$M = z*\frac{\sigma}{\sqrt{n}}$

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

How many days do they need to sample

We need to sample n days.

n is found when M = 300.

We have that $\sigma = 1010$

So

$M = z*\frac{\sigma}{\sqrt{n}}$

$300 = 2.575*\frac{1010}{\sqrt{n}}$

$300\sqrt{n} = 2.575*1010$

$\sqrt{n} = \frac{2.575*1010}{300}$

$(\sqrt{n})^{2} = (\frac{2.575*1010}{300})^{2}$

$n = 75.1$

Rounding up

The need to sample 76 days.