Chapter 2: Exploring Data
Part 1 – Density Curves, Normal Curve, and Assessing Normality
Example #1
Using the following uniform density curve, answer the question:
a. Verify that this is a density curve.
b. What is the probability that the random variable has a value less than 3?
c. What is the probability that the random variable has a value between 3 and 5?
d. What is the percentile for the variable that has a value of 6?
e. What value for the variable is in the 25th percentile?
Example #2
A group of 78 third-grade students in a Midwestern elementary school took a “self-concept” test
that measured how well they felt about themselves. Higher scores indicate more positive selfconcepts. A histogram for these students’ self-concept scores are given below. Draw an
appropriate density curve for summarizing the histogram on the graph above. How would you
describe the shape of this density curve?
Frequency
20
10
0
15
25
35
45
55
65
75
85
SelfConcept
Example #3
Label A, B, and C as either the mean, median, or mode for each picture.
Example #4
For the density curve below, which of the following is true?
a.
b.
c.
d.
e.
The mean and median are equal.
The mean is greater than the median.
The mean is less than the median
The mean could be either greater than or less than the median
The mean is 0.5
The 68-95-99.7 Rule:
Example #5
Find the area between each of the deviations.
Example #6
The mean GPA for Taft HS is 2.9 with the standard deviation of 0.5. What GPA score will place
a student in the top 16% of the class?
Example #7
According to Nielsen Media Research, people watch television an average of 7 hours per day.
Assume that these times are normally distributed with a standard deviation of 4 hours. What
value would place you in the 84th percentile?
Example #9: Here are the lengths in feet of 44 great white sharks:
18.7
16.7
14.3
16.8
12.3
17.8
16.6
13.6
18.6
16.2
9.4
13.2
16.4
12.6
18.2
15.7
15.7
17.8
13.2
19.7
18.3
13.8
13.6
18.7
14.6
12.2
15.3
13.2
15.8
15.2
16.1
16.8
14.9
14.7
13.5
17.6
12.4
19.1
12.1
13.2
16.2
16.4
15.8
22.8
a. Draw a histogram of the data using your calculator. Describe the distribution of the
lengths of the sharks.
b. Compare the mean and the median. Does this comparison support your assessment of the
shape of the distribution in a? Explain.
c. Make a Normal Probability Plot. What does the graph tell you?
Example #10
A Normal probability plot for the amount of lactic acid in a sample of 30 pieces of cheese is
shown below. Is the lactic acid distribution approximately Normal? Justify your answer.
Example #11
The plot shown at the right is a Normal probability plot for a set of data. The data value is
plotted on the x axis, and the standardized value is plotted on the y axis. Which statement is true
for these data?
(a) The data are clearly Normally distributed.
(b) The data are approximately Normally distributed.
(c) The data are clearly skewed to the left.
(d) The data are clearly skewed to the right.
(e) There is insufficient information to determine the shape of the
distribution.
Part 2 – Standardizing and Z-scores
Example #1
The distribution of the duration of human pregnancies (i.e. the number of days between
conception and birth) has been found to be approximately normal with mean  = 266 and the  =
16.
a. What is the Z-Score for a human pregnancy of X = 266?
b. What is the Z-Score for a human pregnancy of X = 250?
Example #2
Let’s us TABLE A to find the following: Draw the picture first, shade the region you want and
look up the X in Table A to find the proportion, probability (percentage) to the left of that zscore. The proportion is also known as probability that the value of a particular member of a
population will fall in the given interval.
a. P (Z < -2.20) =
b. P (Z > 2.20) =
c. P ( Z > -0.95) =
d. P (Z < 1.25) =
e. P (-1.04 < Z < 3.01) =
f. P (0.15 < Z < 1.41) =
Example #3
Suppose that the fuel efficiency (in miles per gallon) of a Beetle varies with each tank of gas
according to a normal distribution with  = 34 and standard deviation  = 3.5 miles per gallon.
a. What proportion of all tanks would get 29 miles per gallon or less?
b. What proportion of all tanks would get 40 miles per gallon or more?
c. What proportion of all tanks would get between 27 and 42 miles per gallon?
d. What proportion of all tanks would get 47 miles per gallon or more? Less than 47
miles per gallon?
Example #4
Golf courses have a wide range of difficulty. Similarly, players differ in ability. In order to
adjust for variations between players, they are often assigned a handicap score. To adjust for
variations between courses, a handicapper decides to compare the golfer’s score against the data
from the course. Suppose course A plays at a mean score of 76 with a standard deviation of 8
strokes with a normal distribution of scores. The mean score for course B is 80 with a standard
deviation of 6 strokes and the scores are normally distributed. If a golfer regularly shoots an 80
on course A, what should be the comparable score on course B?
Inverse Normal Probability Calculations: Un-standardizing
**Sometimes the proportion or percentage is given and you must find the corresponding z-score
and un-standardize the value by finding the X-value.
Example #5: What value(s) of Z cut off the region described?
a. The lowest 11%
b. The highest 30%
c. The highest 7%
d. The middle 50%
Steps for the Inverse Probability Calculation:
 Draw a picture
 Identify the z-value from the given value of the proportion – look up the proportion in
the MIDDLE of Table A.
 Solve for x:
Example # 6
A British company called Molebegon removes unwanted moles from gardens. In 1995, the
European Union announced that the tiny moles are just too difficult to catch. They will not
attempt to catch the smallest 10%. Molebegon’s past records indicate that weights of moles are
normally distributed with a mean of 150 grams and a standard deviation of 32 grams. What is
the cut off weight for the moles they will catch?