Odwalla’s OJ is packaged in $250$ ml bottles and has a process standard deviation of $10$ ml.
In monitoring the fill process, $6$ samples (of $25$ bottles each) were collected and averaged:
What is the total number of bottles sampled? $150$
What is the standard deviation of the distribution of sample means? $2$ ml
The correct answer is supposedly $2$ ml but I keep getting a decimal answer.
Can anyone help, please?
On
Hint $\overline{x}_1 = 249, \overline{x}_2 = 252, \overline{x}_3 = 253, \overline{x}_4 = 248,\overline{x}_5 = 245,\overline{x}_6 = 253 \Rightarrow m = \dfrac{\overline{x}_1+\overline{x}_2+\cdots +\overline{x}_6}{6} = 250 \Rightarrow s_{\overline{x}} = \sqrt{\dfrac{\displaystyle \sum_{k=1}^6 \left(\overline{x}_k-250\right)^2}{6-1}}$
The second part of the question is asking the following: if the standard deviation of a single bottle is $10$ ml, then what is the standard deviation of the average of $25$ bottles? Intuitively, the average of $25$ bottles will tend to have less variability than the variability of a single bottle, because the individual variabilities of the bottles will tend to cancel each other out. The correct formula to apply here is $$\sigma_{\bar x} = \sigma_x/\sqrt{n},$$ where $n$ is the sample size, $\sigma_x$ is the standard deviation of a single measurement, and $\sigma_{\bar x}$ is the standard deviation of the average of $n$ such measurements.