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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:

  • $249$
  • $252$
  • $253$
  • $248$
  • $245$
  • $253$

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?

barak manos
  • 43,109
Little Black Dog
  • 161

2 Answers2

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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.

heropup
  • 135,869
  • Thanks, I understand the intuition part but when I use the (standard error) formula I get: =10/sqrt(150) = 0.816... instead of 2? – Little Black Dog Dec 16 '14 at 12:16
  • $n \ne 150$ in your calculation. It is $n = 25$, because the question is asking for the standard deviation of the distribution of the sample means. You are told that a "sample" in this case constitutes a collection of $25$ bottles, and you have $6$ such samples. So your sample size is $25$, not $150$. – heropup Dec 16 '14 at 12:17
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Hint $\overline{x}_1 = 249, \overline{x}_2 = 252, \overline{x}_3 = 253, \overline{x}_4 = 248,\overline{x}_5 = 245,\overline{x}_6 = 253 \Rightarrow m = \dfrac{\overline{x}_1+\overline{x}_2+\cdots +\overline{x}_6}{6} = 250 \Rightarrow s_{\overline{x}} = \sqrt{\dfrac{\displaystyle \sum_{k=1}^6 \left(\overline{x}_k-250\right)^2}{6-1}}$

DeepSea
  • 77,651
  • The question isn't asking for an estimate based on the data collected. It is asking for the standard deviation of the sampling distribution. Those are two different things. – heropup Dec 16 '14 at 12:23