Question

A coal company wants to create a 90% confidence interval estimate for the mean daily tonnage of coal that they mine. The company reports that the daily output of coal ranges from 1200 to 1800 tons.

A) What is the planning value for the population standard deviation (σ)?

B) If the company desires a margin of error of 40 tons, what should the sample size be?

C) If the company would like the margin of error to be only 30 tons, will the sample size need to increase or decrease? Why?

Answer 1

Solution :

a) According to range rule of thumb, the standard deviation is about one fourth of the range.

Given that, daily output of coal ranges from 1200 to 1800 tons.

i.e. range = 1800 - 1200 = 600

Hence, SD = range/4

SD = 600/4 = 150

The population standard deviation (σ) is 150.

B) The sample size needed to estimate the population mean with 90% confidence level is given as follows :

Where, n is sample size, E is Margin of error, σ is population standard deviation and Z(0.10/2) is critical z-value to construct 90% confidence interval.

We have, E = 40,  σ = 150

Using Z-table we get, Z(0.10/2) = 1.645

Hence, required sample size is,

n = 38

Hence, the sample size should be 38.

c) The formula for the sample size is given in part (b). In the formula we can see that the margin of error is in the denominator. Hence, when we decrease the margin of error, the sample size will increase.

Hence, to reduce the margin of error at 30, the sample size need to be increased.