Download The Normal Proba

STAT 220
Basic Statistics
Autumn 2011
The Normal Probability Curve
Name:
Section:
1. Suppose that body temperature for adults has a normal distribution with a mean of
98.2 F and a S.D. of 0.8 F.
(a) What is the probability that a randomly selected adult has a body temperature
less than or equal to 98.6 F. ?
We can draw a picture and see that we want the area below 98.6 for a normal
curve that is centered at 98.2 with a standard deviation of 0.8. Due to a property
of the normal distribution, this area is equal to the area below a z-score of (98.6 −
98.2)/0.8 = 0.5 on a standard normal curve. We can compute this area in many
different ways. One way is to subtract the area above a z-score of 1/2 from the
total of 1. The area above a z-score of 1/2 is (1-0.3829)/2 which equals 0.3085.
Thus, the area below a z-score of 1/2 is 0.6915.
(b) What is the probability that a randomly selected adult has a body temperature
greater than 99.2 F.?
Using similar arguments as above, the area we seek is above a z-score of 1.25 on a
standard normal curve. The area between ± 1.25 is 0.7887. Hence the area above
1.25 is half of (1-0.7887) which equals 0.1056.
(c) What is the probability that a randomly selected adult has a body temperature
between 97 and 98 degrees Farenheit?
This is the area below a z-score of -3/2 subtracted from the area below a z-score
of -1/4. Manipulating the numbers in the table, the area below -3/2 is 0.0668 and
the area below -1/4 is 0.4013. So the answer is 0.3345.
(d) What is the 95th percentile of the body temperature distribution?
This requires finding the z-score corresponding to a given area of 0.95. From the
table, a z-score of 1.65 has an area of 0.95 below it. This z-score can converted
back to the body temperature distribution by 1.65 ×0.8+98.2 which equals 99.52.
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2. In education, grading on a curve refers to the practice of using a pre-determined distribution of grades among the students in the class. Suppose the normal curve will be
used to grade the 176 students registered for STAT 220 this quarter. Draw a picture of
the distribution of the G.P.A.s, indicating the cut-off points for the middle 68%, 95%
and 99.7% of the scores. What do you like about this system of grading? What don’t
you like?
We rank all the overall scores first and then divide them into bins with the top 2.5% in
the first , 13.5% in the second, 34% in the third, 34% in the fourth, 13.5% in the fifth
and 2.5% in the last bin. We then pick the cut-off GPAs for each of the bins. Usually
the mean and S.D. are specified by the department in which case the cut-off between
the third and fourth bins is the mean and the next cut-off in either direction is ± one
standard deviation. Individuals within each bin are assigned to a GPA based on areas
under the normal table as explained in class.
3. (Adapted from “Elementary Statistics” by Nancy Pfenning) “Remains found of downsized human species” reports that “once upon a time on a tropical island mid-way
between Asia and Australia, there lived a race of little people whose adults stood just
3.5 feet high.” Suppose the height of the smallest race of modern human pygmies
follows a normal distribution with mean 54 inches and S.D. of 2 inches. Do you think
this find warrants the identification of a new human species?
The find corresponds to a z-score of (42-54)/2 which equals -6. This is definitely an
extreme value relative to the height distribution of modern pygmies. It does seem
to suggest that the find may be an observation from a different height distribution,
perhaps a so far undiscovered human species. However, I would hesitate to base that
conclusion only on statistical probabilities because improbable does not mean impossible.
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