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. = $$\frac{\bar X - \mu}{\frac{\sigma}{\sqrt{n} } }$$ ~ N(0,1)where, $$\bar X$$ = sample mean weighing of specimen = $$\frac{3.412+3.416+3.414}{3}$$ = 3.414 g             $$\sigma$$ = population standard deviation = 0.001 g             n = sample of specimen = 3             $$\mu$$ = 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, $$\mu$$ 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 < $$\frac{\bar X - \mu}{\frac{\sigma}{\sqrt{n} } }$$ < 2.5758) = 0.99P( $$-2.5758 \times {\frac{\sigma}{\sqrt{n} } }$$ < $${\bar X - \mu}$$ < $$2.5758 \times {\frac{\sigma}{\sqrt{n} } }$$ ) = 0.99P( $$\bar X-2.5758 \times {\frac{\sigma}{\sqrt{n} } }$$ < $$\mu$$ < $$\bar X+2.5758 \times {\frac{\sigma}{\sqrt{n} } }$$ ) = 0.9999% confidence interval for $$\mu$$ = [ $$\bar X-2.5758 \times {\frac{\sigma}{\sqrt{n} } }$$ , $$\bar X+2.5758 \times {\frac{\sigma}{\sqrt{n} } }$$ ]                                              = [ $$3.414-2.5758 \times {\frac{0.001}{\sqrt{3} } }$$ , $$3.414+2.5758 \times {\frac{0.001}{\sqrt{3} } }$$ ]                                              = [3.4125 , 3.4155]Therefore, 99% confidence interval for this specimen is [3.4125 , 3.4155].