Download Stat 1: Practice Normal Computations with Body Temperature

Stat 1: Practice Normal Computations with Body Temperature
Suppose that the body temperatures of healthy adults vary from person to person according to
a Normal distribution with mean µ = 36.8 and standard deviation σ = 0.4 degrees Celsius. For
brevity we can write
1) Between about what two values do the middle 68% of Celsius body temperatures fall?
Using the 68/95/99.7 rule, we know that about 68% of people have body temperatures within
±1 standard deviation of the mean, so between 36.8−0.4 = 36.4 degrees and 36.8+0.4 = 37.2
degrees Celsius.
2) Between about what two values do the middle 95% of Celsius body temperatures fall?
Again using the 68/95/99.7 rule, we know that about 95% of people have body temperatures
within ±2 standard deviation of the mean, so between 36.8 − 2(0.4) = 36.0 degrees and
36.8 + 2(0.4) = 37.6 degrees Celsius.
3) About what percent of body temperatures are above 37.6 degrees Celsius?
37.6 is 2 standard deviations above the mean. We know that about 95% of people have
temperatures within ±2 sd’s of the mean, so that leaves a total of about 5% with values more
extreme than this (in either direction). The symmetry of the Normal distribution implies
that half of these, or about 2.5% of all people have values more than 2 sd’s above the mean
or above 37.6 degrees (and about 2.5% have values less than 2 sd’s below the mean, or below
36.0 degrees).
4) About what percent are between 36.4 and 37.6 degrees Celsius?
37.6 is two standard deviations above the mean, so only about 2.5% of people have temperatures higher than this and 97.5% have values less than or equal to 37.6. 36.4 is one standard
deviation below the mean, so about 16% (half of the 32% that are not within ±1 standard
deviations of the mean) are below this value. That means that about 97.5% − 16% = 81.5%
of people have temperatures between 36.4 and 37.6 degrees Celsius.
5) Above about what value are the highest 16% of body temperatures?
About 32% of people have temperatures more than one standard deviation from the mean
and, due to the symmetry of the distribution, half of these people, or 16% of all people have
temperature more than one standard deviation above the mean, or 36.8 + 0.4 = 37.2 degrees.
6) About what percent of body temperatures are below 37.0 degrees C (98.6 F)?
The standard score for x = 37.0 is z = 37.0−36.8
= 0.2
0.4
0.4 = 0.5, meaning that 37.0 degrees is 0.5
standard deviations above the mean. Using Table B we find that 69.15% of values from a
Normal distribution are less than 0.5 standard deviations above the mean.
7) What percent are less than 36.2 degrees Celsius?
The standard score for x = 36.2 is z = 36.2−36.8
= −1.5. From Table B we find that 6.68%
0.4
of values are less than 1.5 standard deviations below the mean.
8) What percent are between 36.2 and 37 degrees Celsius?
Subtracting the answers from the last two problems we find that about 69.15−6.68 = 62.47%
of people have temperatures between these two values.
9) Above what value are the highest 10% of temperatures?
We need to find a standard score that corresponds to the 90th percentile (leaving 10% of values
above this threshold). The closest percentile to 90% in Table B is 90.32, which corresponds
to a standard score of z = 1.3. The Celsius temperature that is 1.3 standard deviations above
the mean is 36.8 + (1.3)0.4 = 37.32.
10) Below what value are the lowest 5% of Celsius temperatures?
We need to find the standard score that corresponds to the 5th percentile. We see that
z = −1.6 corresponds to the 5.48th percentile and z = −1.7 corresponds to the 4.46th
percentile. These percentiles are almost equally far from 5%, so we can average them to
get z = −1.65. The Celsius temperature that is 1.65 standard deviations below the mean
is 36.8 − 1.65(0.4) = 36.14. If you didn’t want to interpolate you could use z = 1.6 as
the standard score closest to the 5th percentile, in which case you would find a value of
36.8 − 1.6(0.4) = 36.16.