Question

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

Answer 1

B. 3.414 +/-.00149

Explanation:

Previous concepts

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".

The margin of error is the range of values below and above the sample statistic in a confidence interval.

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

`\bar X` represent the sample mean for the sample

`\mu` population mean (variable of interest)

`\sigma=0.001` represent the population standard deviation known

n=3 represent the sample size

Confidence interval

Since the population standard deviation is known, the confidence interval for the mean is given by the following formula:

`\bar X \pm z_(\alpha/2)(\sigma)/(√(n))` (1)

In order to calculate the mean we can use the following formula:

`\bar X= \sum_(i=1)^n (x_i)/(n)` (2)

The mean calculated for this case is
`\bar X=3.414`

Since the Confidence is 0.99 or 99%, the value of
`\alpha=0.01` and
`\alpha/2 =0.005`, and we can use excel, a calculator or a tabel to find the critical value. The excel command would be: "=-NORM.INV(0.005,0,1)".And we see that
`z_(\alpha/2)=2.58`

Now we have everything in order to replace into formula (1):

`3.414-2.58(0.001)/(√(3))=3.414 -0.00149=3.413`

`3.414+2.58(0.001)/(√(3))=3.414+0.00149=3.415`

So on this case the 99% confidence interval would be given by (3.413;3.415) The best option for this case is:

B. 3.414 +/-.00149