Question

An article contained the following observations on degree of polymerization for paper specimens for which viscosity times concentration fell in a certain middle range:

420 425 427 427 432 433 434 437 439 446 447 448 453 454 465 469
Suppose the sample is from a normal population.
(a) Calculate a 95% confidence interval for the population mean, and interpret it.
(b) Calculate a 95% upper confidence bound for the population mean, and interpret it.

Answer 1

(a) A 95% confidence interval for the population mean is [433.36 , 448.64].

(b) A 95% upper confidence bound for the population mean is 448.64.

Explanation:

We are given that article contained the following observations on degrees of polymerization for paper specimens for which viscosity times concentration fell in a certain middle range:

420, 425, 427, 427, 432, 433, 434, 437, 439, 446, 447, 448, 453, 454, 465, 469.

Firstly, the pivotal quantity for finding the confidence interval for the population mean is given by;

P.Q. =
`(\bar X-\mu)/((s)/(√(n) ) )` ~
`t_n_-_1`

where,
`\bar X` = sample mean =
`(\sum X)/(n)` = 441

s = sample standard deviation =
`\sqrt{(\sum (X-\bar X)^(2) )/(n-1) }` = 14.34

n = sample size = 16

`\mu` = population mean

Here for constructing a 95% confidence interval we have used One-sample t-test statistics as we don't know about population standard deviation.

So, 95% confidence interval for the population mean,
`\mu` is ;

P(-2.131 <
`t_1_5` < 2.131) = 0.95 {As the critical value of t at 15 degrees of

freedom are -2.131 & 2.131 with P = 2.5%}

P(-2.131 <
`(\bar X-\mu)/((s)/(√(n) ) )` < 2.131) = 0.95

P(
`-2.131 * {(s)/(√(n) ) }` <
`{\bar X-\mu}` <
`2.131 * {(s)/(√(n) ) }` ) = 0.95

P(
`\bar X-2.131 * {(s)/(√(n) ) }` <
`\mu` <
`\bar X+2.131 * {(s)/(√(n) ) }` ) = 0.95

95% confidence interval for
`\mu` = [
`\bar X -2.131 * {(s)/(√(n) ) }` ,
`\bar X +2.131 * {(s)/(√(n) ) }` ]

= [
`441-2.131 * {(14.34)/(√(16) ) }` ,
`441+2.131 * {(14.34)/(√(16) ) }` ]

= [433.36 , 448.64]

(a) Therefore, a 95% confidence interval for the population mean is [433.36 , 448.64].

The interpretation of the above interval is that we are 95% confident that the population mean will lie between 433.36 and 448.64.

(b) A 95% upper confidence bound for the population mean is 448.64 which means that we are 95% confident that the population mean will not be more than 448.64.