Question

Assume that the helium porosity (in percentage) of coal samples taken from any particular seam is normally distributed with true standard deviation 0.77.

(a) Compute a 95% CI for the true average porosity of a certain seam if the average porosity for 24 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 10 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 0.49? (Round your answer up to the nearest whole number.)............................. specimens

(d) What sample size is necessary to estimate true average porosity to within 0.23 with 99% confidence? (Round your answer up to the nearest whole number.) .....................specimens

You may need to use the appropriate table in the Appendix of Tables to answer this question.

Answer 1

(a) Though population standard deviation is given , we shall use t distribution as sample size is small.

95% confidence interval is

At 95% confidence with 23 degrees of freedom

tc = 2.0639

Therefore , 95% confidence interval is

=

= ( 4.53 , 5.17)

(b)

Though population standard deviation is given , we shall use t distribution as sample size is small.

98% confidence interval is

At 98% confidence with 9 degrees of freedom

tc = 2.0639

Therefore , 98% confidence interval is

=

= ( 3.87 , 5.25)

In the above two problems we have used population standard deviation in place of s

(c) Margin of error = 0.49

For 95% confidence zc =1.96

Therefore sample size required =10

(d)  

Margin of error = 0.23

For 99% confidence zc =2.58

Therefore sample size required =75