Question

In: Math

Assume that the helium porosity (in percentage) of coal samples taken from any particular seam is...

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?

Solutions

Expert Solution

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


Related Solutions

Assume that the helium porosity (in percentage) of coal samples taken from any particular seam is...
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...
Assume that the helium porosity (in percentage) of coal samples taken from any particular seam is...
Assume that the helium porosity (in percentage) of coal samples taken from any particular seam is normally distributed. Compute a 90% CI for the true average porosity of a certain seam if the average porosity for 50 specimens from the seam was 4.85 with a sample standard deviation of .65.
Assume that the helium porosity (in percentage) of coal samples taken from any particular seam is...
Assume that the helium porosity (in percentage) of coal samples taken from any particular seam is normally distributed with true standard deviation 0.76. (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...
Assume that the helium porosity (in percentage) of coal samples taken from any particular seam is...
Assume that the helium porosity (in percentage) of coal samples taken from any particular seam is normally distributed with true standard deviation 0.78. (a) Compute a 95% CI for the true average porosity of a certain seam if the average porosity for 19 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....
Assume that the helium porosity (in percentage) of coal samples taken from any particular seam is...
Assume that the helium porosity (in percentage) of coal samples taken from any particular seam is normally distributed with true standard deviation 0.78. (a) Compute a 95% CI for the true average porosity of a certain seam if the average porosity for 19 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 13 specimens with a sample average porosity of 4.56....
Assume that the helium porosity (in percentage) of coal samples taken from any particular seam is...
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....
Assume that the helium porosity (in percentage) of coal samples taken from any particular seam is...
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 23 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 13 specimens with a sample average porosity of 4.56. (Round...
Assume that the helium porosity (in percentage) of coal samples taken from any particular seam is...
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...
Assume that the helium porosity (in percentage) of coal samples taken from any particular seam is...
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 16 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 12 specimens with a sample average porosity of 4.56....
Assume that the helium porosity (in percentage) of coal samples taken from any particular seam is...
Assume that the helium porosity (in percentage) of coal samples taken from any particular seam is normally distributed with true standard deviation 0.77. (c) How large a sample size is necessary if the width of the 95% interval is to be 0.42? (Round your answer up to the nearest whole number.) (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.)
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT