Question

Answer ASAP: Probability and Statistics question

Assume that the helium porosity (in percentage) of coal samples taken from any particular seam is normally distributed, but neither the mean LaTeX: \muμ nor the variance LaTeX: \sigma^2σ 2 is known. Suppose that we are given a sample of 25 specimens from the seam, with average porosity 8.85 and standard deviation 0.75.

Answer the following questions:

(a) Construct a 98% confidence interval for LaTeX: \muμ. How large is the margin of error?

(b) Does the interval in part (a) contain the true value of LaTeX: \mu μ? Explain why.

(c) Would a 96% confidence interval based on the same sample be narrower or wider than the 98% confidence interval? You do not need to calculate it but be sure to explain your reasoning clearly.

Answer 1

(a)

Standard error of mean , SE =   = = 0.15

Degree of freedom = n-1 = 15-1 = 14

t value for 98% confidence interval and df = 14 is  2.62

Margin of error = t * SE = 2.62 * 0.15 = 0.393

98% confidence interval for is,

(8.85 - 0.393 , 8.85 + 0.393)

(8.457 ,  9.243 )

(b)

It is not necessary that the interval in part (a) contain the true value of . We are 98% confident that the interval in part (a) contain the true value of .

(c)

With decrease in confidence interval, the critical value of t decreases and consequently, the margin of error and width of the confidence interval decreases. Thus, the 96% confidence interval based on the same sample will be narrower than the 98% confidence interval.