Question

Please use the accompanying Excel data set or accompanying Text file data set when completing the following exercise. An article in Technometrics (1999, Vol. 41, pp. 202-211) studied the capability of a gauge by measuring the weight of paper. The data for repeated measurements of one sheet of paper are in the following table. Construct a 90% one-sided, upper confidence interval for the standard deviation of these measurements. Assume population is approximately normally distributed. Round your answer to 3 decimal places. Observations 3.481 3.448 3.485 3.475 3.472 3.477 3.472 3.464 3.472 3.470 3.470 3.470 3.477 3.473 3.474 os

Answer 1

SampleVariance (ti – 7) = n- - 1

$=\frac{1}{15-1}\times 0.000966=\frac{0.000966}{14}=0.000069$

The 90% confidence interval for the variance is given by,

СІ (n-1)s XR <a < (n-1)s XL 2 2

The one sided upper confidence interval can be obtained by computing,

$\sigma ^2\leq \frac{(n-1)s^2}{\chi _L^2}$

where, n= sample size=15

s²= sample variance=0.000069

Kevin 2 Scanned with CamScanner

From chi square table with (n-1)=14 degrees of freedom and $\alpha =0.95$ ,

$\chi _L^2=6.571$

$\therefore$ upper confidence interval is given by,

$\sigma ^2\leq \frac{14\times 0.000069}{6.571}$

$=\frac{0.000966}{6.571}=0.000147$

This is the upper confidence interval for the variance.

Hence, the upper confidence interval for the standard deviation is obtained by taking the square root of upper confidence interval for the variance.

$\therefore$ The 90% confidence interval for the standard deviation is given by,

$\sigma \leq 0.01212$

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