Cem Hayes

2020-12-24

Juan makes a measurement in a chemistry laboratory and records the result in his lab report. The stardard deviation of lab measurements made by students is $\sigma =10$ milligrams. Juan repeats the measurement 3 times and records the mean xbar of his 3 measurements.
(a) What is the standard deviation of Juan's mean result? (That is, if Juan kept making sets of 3 measurements and averaging them, what would be the standard deviation of all his xbar's?)
(b) How many times must juan repeat the measurement to reduce the standard deviation of xbar to5? Explain to someone who knows nothing about statistics the advantage of reporting the average of several measurements rather than the result of a single measurement.

Clara Reese

Step 1
Given,
The standard deviation of lab measurements made by students is $\sigma =10$ milligrams.
Sample size is $n=3$
Now, the standard deviation of the sampling distribution is given by,
${\sigma }_{\stackrel{―}{x}}=\frac{\sigma }{\sqrt{n}}$
Step 2
a)
${\sigma }_{\stackrel{―}{x}}=\frac{\sigma }{\sqrt{n}}$
$=\frac{10}{\sqrt{3}}=5.7735$
b)
${\sigma }_{\stackrel{―}{x}}=\frac{\sigma }{\sqrt{n}}$
$5=\frac{10}{\sqrt{n}}$
$\sqrt{n}=\frac{10}{5}$
$\sqrt{n}=2⇒n=4$
Juan must repeat the measurement 4 times to reduce the standard deviation of xbar to5
Reporting the average of several measurements has more advantage than the result of a single measurement. Since, the average of several measurements is more likely to be closer to the mean, than the result of a single measurement.

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