Question

how large of a sample is needed to estimate the mean of a normally distributed population for each of the following?a. ME= 5, sigma=40, alpha=0.01b. ME=10; sigma=40; alpha=0.01c. compare and comment on your answers to parts a and b.

Answer 1

Answer:

a: 425

b: 107

c: read below

Step-by-step explanation:

Assuming ME means Error:

We have:

n = to be determined

ME = 5

σ = 40

α = 0.01

The z-value we will use for α = 0.01 is: 2.575

See attached photo 1 for the calculation of n

For the second situation, it is calculated the same way, we just have E = 10

See second attached photo for the calculation of this value of n...

For part c:  The sample size is much smaller for the second situation because we are allowing more error in our estimate, so we can be less accurate.  We don't need as large of a sample size if we don't want to be as accurate

Answer 2

Using the z-distribution, it is found that:

a. A sample of 425 is needed.

b. A sample of 107 is needed.

c. The margin of error is inversely proportional to the square root of the sample size, hence, as the margin of error increased, the sample size can decrease.

The margin of error of the z-distribution, considering a critical value of z, a population standard deviation of $\sigma$ and a sample size of n is given by:

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

For this problem, $\alpha = 0.01$, hence the critical value is z with a p-value of $1 - \frac{\alpha}{2} = 1 - 0.005 = 0.995$, hence z = 2.575.

Also, the population standard deviation is $\sigma = 40$

Item a:

The sample size is n for which M = 5, hence:

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

$5 = 2.575\frac{40}{\sqrt{n}}$

$5\sqrt{n} = 2.575(40)$

Dividing both sides by 5:

$\sqrt{n} = 2.575(8)$

$(\sqrt{n})^2 = [2.575(8)]^2$

$n = 424.36$

Rounding up:

A sample of 425 is needed.

Item b:

The sample size is n for which M = 10, hence:

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

$10 = 2.575\frac{40}{\sqrt{n}}$

$10\sqrt{n} = 2.575(40)$

Dividing both sides by 10:

$\sqrt{n} = 2.575(4)$

$(\sqrt{n})^2 = [2.575(4)]^2$

$n = 106.1$

Rounding up, a sample of 107 is needed.

Item c:

The margin of error is inversely proportional to the square root of the sample size, hence, as the margin of error increased, the sample size can decrease.

A similar problem is given at https://brainly.com/question/18297380