Download Normal Curve Calculations The Empirical Rule that we have applied

Normal Curve Calculations
The Empirical Rule that we have applied to bell-shaped distributions is based on a special
mathematical distribution called the normal distribution. The normal distribution is
symmetric about the mean µ . The standard deviation σ describes the spread from the
mean. See figure 1 below. In fact the Empirical Rule says that about 68% of the data
falls in the interval µ ± σ and that about 95% falls in the interval µ ± 2σ.
µ−σ µ
Figure 1
µ+σ
288 401 514 627 740
Figure 2
In the year 2000 the SAT math scores had a mean of 514 and a standard deviation of 113.
The distribution was approximated well by a normal distribution. The sketch is shown in
figure 2 above. By the Empirical Rule we can say that 68% of the SAT math scores were
between 401 and 627, and that 95% of the scores were between 288 and 740.
What if we wanted to know the percentage of the scores that were between 500 and 700?
Or what was the percentage above 700 ? These are the type of questions that we will
learn to answer now. The key to answering these questions is to ask how many standard
deviations are these scores from the mean. For example how many standard deviations is
700 from the mean of 514 ? (try to answer this before reading further).
First 700 is 700 – 514 = 186 points above the mean. How many standard deviations are
in 186 points? Answer: 186/113 = 1.65 standard deviations. This is called the
standardized score or z-score. The calculation can be summarized by the formula
z=
€
x −µ
value − mean
or z − score =
σ
stan dard deviation
Let’s return to the problem of determining the percentage of the scores that fall between
500 and 700. First calculate the z-scores for each of the scores 500 and 700. The
calculations are shown in the sketch in Figure 3. After finding the z-scores we need to
determine the percentage of SAT scores between 500 and 700. To do this we use a
command in the TI-83 calculator. The command is normalcdf and is found under the
DISTR menu which is accessed by 2nd VARS. Figure 4 shows the shaded area
corresponding to the desired percentage. The conclusion is that 49.83% of the SAT
math scores were between 500 and 700.
normalcdf(-.12,1.65) = .4983
288
401 514 627 740
500 − 514
700 − 514
z=
z=
113
113
z ≈ −.12
z ≈ 1.65
Figure 3
288 401 514 627 740
Figure 4
€Exercise 1: What
€ percentage of the scores were between 600 and 700 ?
(Answer 17.3%)
Example 2. What percentage of the scores are higher than 700 ?
Step 1. Make sketch
Step 2. Calculate z-score
Step 3. Use normalcdf
288 401 514 627 740
700 − 514
z=
113
z ≈ 1.65
normalcdf(1.65,100) = .0495
Therefore we conclude that about 4.95% of the SAT math scores were higher than 700.
Note: Since there is no right boundary on
the shaded region we use 100 for the z-score
on the right. Any large number will give
about the same answer.
€
Exercise 2: What percentage of the scores were below 600?
(Answer: 77.67% )
The diagram below summarizes the procedure for finding the percentage of a normal
distribution between two x values.
x values
z-scores
area
In the next example we will need to reverse this process.
Example 3 Suppose we want to know the SAT math score that has 25% of the scores below
it.
First make the sketch. The SAT score we are
looking for is indicated by the x.
We will first find the z-score that
corresponds to the score x. To do this we
use the calculator command InvNorm.
25%
288 401 514 627 740
x
The InvNorm command needs to be told the area to the left of the score we want. In
this example we use InvNorm(0.25) = -0.67. This the z-score that corresponds to the
SAT score, x, that has 25% of the scores below it. To find x we just solve as shown
below.
x − 514
113
−.67(113) = x − 514
−.67(113) + 514 = x
x ≈ 438
Therefore a score of 438 on the SAT math test was higher than 25% of all the SAT
math scores. Note that this means that 438 is the lower quartile of the SAT math
scores in the year 2000.
−.67 =
€
Problems
1. Based on the data from an article in Consumer Reports, the length of time CD players
will work before replacement is needed is normally distributed with a mean of 7.1
years and a standard deviation of 1.4 years.
A) Draw a sketch of the distribution labeling the mean and two standard deviations in
each direction.
B) According to the Empirical Rule, 95% of the CD players will last between _______
years and ________ years before they need to be replaced.
C) What percent of CD players will last between 5 and 8 years before needing
replacement?
D) Joe is deciding if he should purchase the 5 year replacement warranty, which will
replace his CD player if it should break within the first five years of ownership.
What percent of CD players will last more than 5 years before needing
replacement? Should he purchase the warranty? Defend your answer.
2. The gas mileage for a certain model car varies from car to car. Suppose that the distribution
of mileages is summarized by a normal distribution with a mean of 26 and standard
deviation of 3 miles per gallon.
(A) Sketch the distribution labeling the mean and two standard deviations in each
direction.
(B) What percentage of the cars have a mileage more than 30 miles per gallon ?
(C) What percentage of cars have mileages between 22 and 30 miles per gallon ?
(D) Almost all of the cars will have mileages between
3.
and
. Explain.
The pulse rate of the adult male population between age 18 and 25 is a normally
distributed variable with mean of 72 and standard deviation of 12. Suppose that
anyone with a pulse rate of 100 or more is not allowed to be a pilot.
A) What percentage of adult males in this age range are not allowed to be pilots because
of a high pulse rate ?
B) What pulse rate would an adult male have in order to have a rate in the slowest 25%
of adult males?
4. The GRE is a test taken by college students who intend to go to graduate school. For all
college seniors and graduates who took the exam in 1989-1992 the scores on the GRE were
normally distributed with a mean of 497 and a standard deviation of 115. Assume this
distribution still applies. If a graduate school wants to admit only those whose score is in
the top 20% of all scores, what would they use as the cutoff score to get in the school ?
Include a sketch and show the cutoff score on the sketch. Write your answer in a complete
sentence.
5. The weights of boxes of Raisin Nut Bran cereal produced at a factory vary from box to box.
Suppose that these weights are normally distributed with a mean of 595 grams and a standard
deviation of 12 grams.
A) Sketch the distribution of weights and label the mean and two standard deviations in
each direction.
B) What is the weight of a box of cereal that is in the lightest 10%?
C) What percentage of the boxes will weigh between 590 and 610 grams ? Include a
sketch and show your work. Write your answer in a complete sentence.
D) The quality control department will only ship a box if it weighs at least 570 grams.
What percentage of the boxes will be shipped? Include a sketch and show your work.
Write your answer in a complete sentence.
6. Referring to problem number 5, suppose the factory can control the mean weight of the boxes
of cereal by adjusting the machine that fills the boxes. If the quality control department will
ship a box only if it weighs at least 570 grams, and they want to discard only 1% of the
boxes, what is the mean weight they should use to adjust the machine? Assume the standard
deviation stays at 12 grams.
7. Suppose the scores on the SAT math test have a mean of 570 and a standard deviation
of 50. The scores on the ACT test have a mean of 17 with a standard deviation of 3.
Assume both are normally distributed.
A) What is the z-score for a score of 700 on the SAT math test? What is the z-score
for a score of 26 on the ACT test?
B) Which is better : 700 on the SAT or a score of 26 on the ACT test. Explain.
8. Suppose that I.Q. scores on a commonly used test are normally distributed with a mean of
100 and a standard deviation of 15 and ths scores are normally distributed. Find the I.Q.
scores that correspond to the lower quartile, the median, and the upper quartile.
Answers
1 B) 4.3 to 9.9 C) 67.3% D) 93.3%
2. B) 9.1% C) 82% D) 17 and 35 mpg
3. 0.98% B) 64 beats per minute 4.) 594
6) 598
7A) SAT: z = 2.6 ACT: z = 3
5B) 579.6 C) 55.6% D) 98%
8) 90,100,110