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Establishing Normality Based on the $5$-Number Summary

APSTAT-BTFV4A

The $5$-number summary for a certain distribution is as follows:

$28$
$53$
$75$
$97$
$122$

The mean of the distribution is $75.10$ and the standard deviation is $31.27$.

Based upon this information, is the distribution approximately normal? Why or why not?

A

Yes, because the $Q3$ value and $Q1$ value are equally far from the median, and the median is approximately equal to the mean.

B

Yes, because the values in the $5$-number summary are completely symmetric around the mean.

C

Yes, because approximately $50\%$ of the data falls within one standard deviation of the mean. This is what would be expected in a normal distribution.

D

No, because $100\%$ of the data falls within $2$ standard deviations of the mean. In a normal distribution, the spread of data would be greater.

E

No, because the data contains a high outlier. In a normal distribution, no outliers would be present.