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

Answer:

B. 3.414 +/-.00149

Step-by-step 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:

$\bar X \pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}$   (1)

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

$\bar X= \sum_{i=1}^n \frac{x_i}{n}$ (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):

$3.414-2.58\frac{0.001}{\sqrt{3}}=3.414 -0.00149=3.413$    

$3.414+2.58\frac{0.001}{\sqrt{3}}=3.414+0.00149=3.415$

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