Question

Assume that the helium porosity (in percentage) of coal samples taken from any particular seam is normally distributed with true standard deviation 0.80. a) Compute a 95% CI for the true average porosity of a certain seam if the average porosity for 15 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 17 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 0.39? (Round your answer up to the nearest whole number.) specimens d) What sample size is necessary to estimate true average porosity to within 0.24 with 99% confidence? (Round your answer up to the nearest whole number.)

Answer 1

a)

sample mean, xbar = 4.85
sample standard deviation, σ = 0.8
sample size, n = 15

Given CI level is 95%, hence α = 1 - 0.95 = 0.05
α/2 = 0.05/2 = 0.025, Zc = Z(α/2) = 1.96

ME = zc * σ/sqrt(n)
ME = 1.96 * 0.8/sqrt(15)
ME = 0.4

CI = (xbar - Zc * s/sqrt(n) , xbar + Zc * s/sqrt(n))
CI = (4.85 - 1.96 * 0.8/sqrt(15) , 4.85 + 1.96 * 0.8/sqrt(15))
CI = (4.45 , 5.25)

b)

sample mean, xbar = 4.56
sample standard deviation, σ = 0.8
sample size, n = 17

Given CI level is 98%, hence α = 1 - 0.98 = 0.02
α/2 = 0.02/2 = 0.01, Zc = Z(α/2) = 2.33

ME = zc * σ/sqrt(n)
ME = 2.33 * 0.8/sqrt(17)
ME = 0.45

CI = (xbar - Zc * s/sqrt(n) , xbar + Zc * s/sqrt(n))
CI = (4.56 - 2.33 * 0.8/sqrt(17) , 4.56 + 2.33 * 0.8/sqrt(17))
CI = (4.11 , 5.01)

c)

The following information is provided,
Significance Level, α = 0.05, Margin or Error, E = 0.195, σ = 0.8

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 * 0.8/0.195)^2
n = 64.66

Therefore, the sample size needed to satisfy the condition n >= 64.66 and it must be an integer number, we conclude that the minimum required sample size is n = 65
Ans : Sample size, n = 65

d)

The following information is provided,
Significance Level, α = 0.01, Margin or Error, E = 0.24, σ = 0.8

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 * 0.8/0.24)^2
n = 73.96

Therefore, the sample size needed to satisfy the condition n >= 73.96 and it must be an integer number, we conclude that the minimum required sample size is n = 74
Ans : Sample size, n = 74