Question

You have a list of 1,000 random, normally distributed numbers with a mean of 20 and a standard deviation of 3. Which statement is true?Approximately 340 numbers lie between 14 and 20.
Approximately 680 numbers lie between 17 and 29.


Approximately 340 numbers lie between 20 and 23.


Approximately 170 numbers lie between 14 and 17.

Answer 1

You're going to have to work out the z-scores of both values in each option and see if it makes sense.

n = m + sz

The mean is 20 and the standard deviation is 3.

So let's find try the z-scores of the outer range values and determine their probabilities:

from 14-20:

n = m + sz
20 = 20 + 3z and 14 = 20 + 3z
0 = 3z and -6 = 3z
z = 0 and z = -2

So now using a z-score table, such as the one below, find the probabilities.

z = 0 is easy, it's 0.5
z = -2 is 0.02275

Subtract the smaller from the larger to get the probability of getting in the range:

0.5 - 0.02275 = 0.47725 (so times 1000, you get 477.25, which is not about 340, so this option is wrong)
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Now trying option 2, from 17-29:

n = m + sz
29 = 20 + 3z and 17 = 20 + 3z
9 = 3z and -3 = 3z
z = 3 and z = -1

0.99865 - 0.158655 = 0.839995 (not 680, so not the answer)
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Now trying option 3, from 20-23:

n = m + sz
23 = 20 + 3z and 20 = 20 + 3z
3 = 3z and 0 = 3z
z = 1 and z = 0

0.841345 - 0.5 = 0.341345 (close to 340, so is your answer)