Statistics for the Behavioral Sciences by Frederick J. Gravetter, Larry B. Wallnau ISBN 10: 1305504917 ISBN 13: 9781305504912

SECTION 6.3 | Probabilities and Proportions for Scores from a Normal Distribution 177
The final step is to transform the z-score into an X value. By definition, a z-score
of +1.28 corresponds to a score that is located above the mean by 1.28 standard deviations.
One standard deviation is equal to 10 points (σ = 10), so 1.28 standard deviations is
1.28 σ = 1.28(10) = 12.8 points
Thus, our score is located above the mean (μ = 24.3) by a distance of 12.8 points.
Therefore,
X = 24.3 + 12.8 = 37.1
The answer for our original question is that you must commute at least 37.1 minutes a
day to be in the top 10% of American commuters.
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EXAMPLE 6.11
Again, the distribution of commuting times for American workers is normal with a mean
of μ = 24.3 minutes and a standard deviation of σ = 10 minutes. For this example, we will
find the range of values that defines the middle 90% of the distribution. The entire distribution
is shown in Figure 6.15 with the middle portion shaded.
The 90% (0.9000) in the middle of the distribution can be split in half with 45% (0.4500)
on each side of the mean. Looking up 0.4500, in column D of the unit normal table, you
will find that the exact proportion is not listed. However, you will find 0.4495 and 0.4505,
which are equally close. Technically, either value is acceptable, but we will use 0.4505 so
that the total area in the middle is at least 90%. Reading across the row, you should find a
z-score of z = 1.65 in column A. Thus, the z-score at the right boundary is z = +1.65 and
the z-score at the left boundary is z = –1.65. In either case, a z-score of 1.65 indicates a
location that is 1.65 standard deviations away from the mean. For the distribution of commuting
times, one standard deviation is σ = 10, so 1.65 standard deviations is a distance of
1.65 σ = 1.65(10) = 16.5 points
Therefore, the score at the right-hand boundary is located above the mean by 16.5 points
and corresponds to X = 24.3 + 16.5 = 40.8. Similarly, the score at the left-hand boundary
is below the mean by 16.5 points and corresponds to X = 24.3 – 16.5 = 7.8. The
middle 90% of the distribution corresponds to values between 7.8 and 40.8. Thus, 90% of
American commuters spend between 7.8 and 40.8 minutes commuting to work each day.
Only 10% of commuters spend either more time or less time.
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Middle 90%
s 5 10
FIGURE 6.15
The distribution of commuting
times for American
workers. The problem is to
find the middle 90% of the
distribution.
7.8 m 5 24.3 40.8
21.65 0
1.65