Question

Assume that the helium porosity (in percentage) of coal samples taken from any particular seam is normally distributed with true standard deviation 0.71.

(a) Compute a 95% CI for the true average porosity of a certain seam if the average porosity for 25 specimens from the seam was 4.85. (Round your answers to two decimal places.) ,

(b) Compute a 98% CI for true average porosity of another seam based on 14 specimens with a sample average porosity of 4.56. (Round your answers to two decimal places.) ,

(c) How large a sample size is necessary if the width of the 95% interval is to be 0.48? (Round your answer up to the nearest whole number.) specimens

(d) What sample size is necessary to estimate true average porosity to within 0.25 with 99% confidence? (Round your answer up to the nearest whole number.) specimens

Answer 1

a) From standard normal tables, we have here:
P(-1.96 < Z < 1.96) = 0.95

Therefore the confidence interval here is obtained as:

this is the required 95% confidence interval for the population mean here.

b) From standard normal tables, we have here:
P(Z < 2.326) = 0.99

Therefore, due to symmetry, we have here:
P( -2.326 < Z < 2.326) = 0.98

Therefore the confidence interval here is obtained as:

This is the required confidence interval here.

c) The width of 0.48 means, a margin of error of 0.48 /2 = 0.24

Therefore, we have here:

Therefore 34 is the minimum sample size required here.

d) From standard normal tables, we have:
P(-2.576 < Z < 2.576) = 0.99

Therefore the sample size here is computed as:

Therefore 54 is the required sample size here.