MATH SOLVE

5 months ago

Q:
# A laboratory scale is known to have a standard deviation (sigma) or 0.001 g in repeated weighings. Scale readings in repeated weighings are Normally distributed with mean equal to the true weight of the specimen. Three weighings of a specimen on this scale give 3.412, 3.416, and 3.414 g. A 99% confidence interval for this specimen is O A. 3.412 +-.00113 B. 3.414 +/-.00149 C. 3.414 +/-.00002 D. 3.414+-00231

Accepted Solution

A:

Answer:99% confidence interval for the given specimen is [3.4125 , 3.4155].Step-by-step explanation:We are given that a laboratory scale is known to have a standard deviation (sigma) or 0.001 g in repeated weighing. Scale readings in repeated weighing are Normally distributed with mean equal to the true weight of the specimen. Three weighing of a specimen on this scale give 3.412, 3.416, and 3.414 g.Firstly, the pivotal quantity for 99% confidence interval for the true mean specimen is given by; P.Q. = [tex]\frac{\bar X - \mu}{\frac{\sigma}{\sqrt{n} } }[/tex] ~ N(0,1)where, [tex]\bar X[/tex] = sample mean weighing of specimen = [tex]\frac{3.412+3.416+3.414}{3}[/tex] = 3.414 g [tex]\sigma[/tex] = population standard deviation = 0.001 g n = sample of specimen = 3 [tex]\mu[/tex] = population meanHere for constructing 99% confidence interval we have used z statistics because we know about population standard deviation (sigma).So, 99% confidence interval for the population mean, [tex]\mu[/tex] is ;P(-2.5758 < N(0,1) < 2.5758) = 0.99 {As the critical value of z at 0.5% level of significance are -2.5758 & 2.5758}P(-2.5758 < [tex]\frac{\bar X - \mu}{\frac{\sigma}{\sqrt{n} } }[/tex] < 2.5758) = 0.99P( [tex]-2.5758 \times {\frac{\sigma}{\sqrt{n} } }[/tex] < [tex]{\bar X - \mu}[/tex] < [tex]2.5758 \times {\frac{\sigma}{\sqrt{n} } }[/tex] ) = 0.99P( [tex]\bar X-2.5758 \times {\frac{\sigma}{\sqrt{n} } }[/tex] < [tex]\mu[/tex] < [tex]\bar X+2.5758 \times {\frac{\sigma}{\sqrt{n} } }[/tex] ) = 0.9999% confidence interval for [tex]\mu[/tex] = [ [tex]\bar X-2.5758 \times {\frac{\sigma}{\sqrt{n} } }[/tex] , [tex]\bar X+2.5758 \times {\frac{\sigma}{\sqrt{n} } }[/tex] ] = [ [tex]3.414-2.5758 \times {\frac{0.001}{\sqrt{3} } }[/tex] , [tex]3.414+2.5758 \times {\frac{0.001}{\sqrt{3} } }[/tex] ] = [3.4125 , 3.4155]Therefore, 99% confidence interval for this specimen is [3.4125 , 3.4155].