Question

Assume that the helium porosity (in percentage) of coal samples taken from any particular seam is normally distributed with true standard deviation 0.74.

(a) Compute a 95% CI for the true average porosity of a certain seam if the average porosity for 20 specimens from the seam was 4.85. 

(b) Compute a 98% CI for true average porosity of another seam based on 13 specimens with a sample average porosity of 4.56.

(c) How large a sample size is necessary if the width of the 95% interval is to be 0.4? 

(d) What sample size is necessary to estimate true average porosity to within 0.21 with 99% confidence?

Answer 1

a)

sample mean, xbar = 4.85
sample standard deviation, s = 0.74
sample size, n = 20
degrees of freedom, df = n - 1 = 19

Given CI level is 95%, hence α = 1 - 0.95 = 0.05
α/2 = 0.05/2 = 0.025, tc = t(α/2, df) = 2.093

ME = tc * s/sqrt(n)
ME = 2.093 * 0.74/sqrt(20)
ME = 0.35

CI = (xbar - tc * s/sqrt(n) , xbar + tc * s/sqrt(n))
CI = (4.85 - 2.093 * 0.74/sqrt(20) , 4.85 + 2.093 * 0.74/sqrt(20))
CI = (4.50 , 5.20)

b)

sample mean, xbar = 4.56
sample standard deviation, s = 0.74
sample size, n = 13
degrees of freedom, df = n - 1 = 12

Given CI level is 98%, hence α = 1 - 0.98 = 0.02
α/2 = 0.02/2 = 0.01, tc = t(α/2, df) = 2.681

ME = tc * s/sqrt(n)
ME = 2.681 * 0.74/sqrt(13)
ME = 0.55

CI = (xbar - tc * s/sqrt(n) , xbar + tc * s/sqrt(n))
CI = (4.56 - 2.681 * 0.74/sqrt(13) , 4.56 + 2.681 * 0.74/sqrt(13))
CI = (4.01 , 5.11)

c)

The following information is provided,
Significance Level, α = 0.05, Margin or Error, E = 0.2, σ = 0.74

The critical value for significance level, α = 0.05 is 1.96.

The following formula is used to compute the minimum sample size required to estimate the population mean μ within the required margin of error:
n >= (zc *σ/E)^2
n = (1.96 * 0.74/0.2)^2
n = 52.59

Therefore, the sample size needed to satisfy the condition n >= 52.59 and it must be an integer number, we conclude that the minimum required sample size is n = 53
Ans : Sample size, n = 53

d)

The following information is provided,
Significance Level, α = 0.01, Margin or Error, E = 0.21, σ = 0.74

The critical value for significance level, α = 0.01 is 2.58.

The following formula is used to compute the minimum sample size required to estimate the population mean μ within the required margin of error:
n >= (zc *σ/E)^2
n = (2.58 * 0.74/0.21)^2
n = 82.65

Therefore, the sample size needed to satisfy the condition n >= 82.65 and it must be an integer number, we conclude that the minimum required sample size is n = 83
Ans : Sample size, n = 83