Download Normal Distribution and Sampling Distributions

Normal Distribution and Sampling Distributions
1. Answer the following questions about an exam with scores that followed a normal distribution.
Round all answers to two decimals. (10 points)
a. A students z-score was found to be -1.45, did this student score higher or lower than the
average score? How do you know this? (2 points)
b. A students z-score was found to be +1, what percentage of students scored higher than
this student? Draw a picture of the curve and correctly identify the area to be found for
full credit. (2 points)
c. Another students z-score was found to be +2, what percentage of students scored lower
than this student? Draw a picture of the curve and correctly identify the area to be
found for full credit. (2 points)
d. Another students z-score was found to be -2, what percentage of students scored lower
than this student? (2 points)
e. Another students z-score was found to be -1, what percentage of students scored lower
than this student? (2 points)
Question 1 tests the students “basic” understanding of z-scores and normal distributions.
2. The distribution of hours worked per week by husbands in the United States in the early 1990’s
had a mean of 42.6 and a standard deviation of 12.9. (7 points)
a. Without actually calculating values of z, tell whether each of these sample means will
have a positive or negative standardized score: 40, 50, 38, 37. Draw a picture of the
curve, and plot all points, for full credit. (5 points)
b. Without actually calculating values of z, tell which one of these sample means will have
the most extreme and which will have the least extreme standardized score: 40, 50, 38,
37. (2 points)
Question 2 tests the students “basic” understanding of z-scores and how they distribute themselves
along a normal curve.
3. Weights of 1-year-old male children are fairly normally distributed with a mean of 26 pounds,
and a standard deviation of 3 pounds. (7 points)
a. Find the Z-score for a male 1-year-old who weighs 20 pounds. (2 points)
b. Find the weight (to the nearest pound) of a 1-year-old male whose z-score is -1.67.
(2 points)
c. What percentage of male 1-year-olds weigh less than 20 pounds? Draw a picture of the
curve and locate area for full credit. Round to two decimal places. (3 points)
Question 3 expands on the students “basic” understanding of z-scores and normal distributions and
applies them to a real life example.
4. A survey found that people keep their television sets an average of 4.8 years. The standard
deviation is 0.89 year. Assume the variable is normally distributed. (175 points)
a. Draw a proper sketch of the curve, label any significant points. (3 points)
b. What is the probability that their TV is less than 4.8 years old? (2 points)
c. What is the probability that their TV is less than 2.5 years? (4 points)
d. What is the probability that their TV is more than 4.2 years? (4 points)
e. What is the probability that their TV is more than 5 years? (4 points)
Question 4 again expands on the students basic knowledge at yet a deeper level. It asks students to
understand the true structure of a normal distribution and what it depicts.
5. The average age of CEO’s is 56 years. Assume the variable is normally distributed. If the
standard deviation is four years, find the probability that the age of a randomly selected CEO
will be in the following range. (9 points)
a. Between 53 and 59 years old. (3 points)
b. Between 58 and 63 years old. (3 points)
c. Between 50 and 55 years old. (3 points)
Question 5 goes into depth about the normal distribution and leads students into a deeper
understanding of the normal distribution and a brief introduction to the next level of the subject.
Two questions I wouldn’t include on my test
1) Suppose you are told the mean and standard deviation for amounts that a sample of students have in
their bank accounts. Why is this not enough to sketch the distribution of amounts?
I do not like this problem because it does not address what we are trying to get across to the students.
The necessary components to draw a curve are not the tasks that we wish to fulfill. It is too broad and
vague.
2) Decide if each of the following variables is most likely to have a shape that is approximately normal,
or skewed left/low outliers, or skewed right/high outliers.
I am not concerned about whether or not my students can identify which variables have a normal
distribution or not. Also I am especially not concerned if my students can identify the skewness of a
graph, this is a skill that they will acquire later in this course or a second stats course.
Solutions
1a) Less than the average (1 point)
Since the z-score is less than 0 (1 point)
1b)
(1 point)
Approximately 15.87% (1 point)
1c)
(1 point)
Approximately 97.73% (1 point)
1d) 100 - 97.73
Approximately 2.27% (2 points)
1e) 15.87% (2 points)
2a)
(1 point)
Negative Score: 37, 38, 40 (3 points)
Positive Score: 50 (1 point)
2b) Most Extreme: 50 (1 point)
Least Extreme: 37 (1 point)
3a)
20−26
3
3b)
𝑥−26
3
=
−6
3
= -2 (2 points)
= -1.67, => x = 20.99 => x = 21 pounds (2 points)
3c)
(1 point)
z-score is -2 (1 point)
percentage less than 20 is approximately 2.28%. (1 point)
4a)
graph: (1 point)
Mean: (1 point)
Mean +/- s.d: (1 point)
4b) 50% (2 points)
4c) z =
2.5−4.8
.89
= -2.58 (2 points)
0.494% are less than 2.5 years (2 points)
4d) z =
4.2−4.8 −.6
= .89
.89
= -.67 (2 points)
74.86% are more than 4.2 years (2 points)
4e) z =
5−4.8
.89
.2
= .89 = .22 (2 points)
41.29% are more than 5 years old (2 points)
5a) z =
z=
59−56 3
=4
4
53−56 −3
= 4
4
= .75 (1 point)
= -.75 (1 point)
77.34 - 22.66 = 54.68% (1 point)
5b) z =
z=
63−56 7
=
4
4
58−56 2
=4
4
= 1.75 (1 point)
= .5 (1 point)
95.99 – 69.15 = 26.84% (1 point)
5c) z =
z=
55−56
4
=
50−56 −4
= 4
4
−1
4
= -.25 (1 point)
= -1 (1 point)
40.13 – 15.87 = 24.26% (1 point)