Question

If the random variable x is normally distributed, ______ percent of all possible observed values of x will be within three standard deviations of the mean. 68.26 95.44 99.73 100 none of these

Answer 1

That would be 99.73 per cent.

Answer 2

99.73

Explanation:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

Percentage within 3 standard deviations of the mean.

pvalue of Z = 3 subtracted by the pvalue of Z = -3.

Z = 3 has a pvalue of 0.9987

Z = -3 has a pvalue of 0.0013

0.9987 - 0.0013 = 0.9974 = 99.74%

A small rounding difference, but the answer is:

99.73