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Naresh Narsing · Answer 1 · 2021-10-24T13:26:11+0000

Answer:

a) 3 standard deviations above 16

b) More than 2 standard deviations of the mean, so yes, 22 inches is faw away from the mean of 16 inches.

c) Less than 2 standard deviations, so not far away.

Explanation:

Z-score:

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.

If Z < -2 or Z > 2, X is considered to be far away from the mean.

In this question, we have that:

$\mu = 16$

(a) Suppose the data come from a sample whose standard deviation is 2 inches. How many standard deviations is 22 inches from 16 inches?

This is Z when
$X = 22, \sigma = 2$ .

So

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

Z = (22 - 16)/(2)

Z = 3

So 22 inches is 3 standard deviations fro 16 inches.

(b) Is 22 inches far away from a mean of 16 inches?

3 standard deviations, more than two, so yes, 22 inches is far away from a mean of 16 inches.

(c) Suppose the standard deviation of the underlying data is 4 inches. Is 22 inches far away from a mean of 16 inches?

Now
$\sigma = 4$

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

Z = (22 - 16)/(4)

Z = 1.5

1.5 standard deviations from the mean, so 22 inches is not far away from the mean.

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