Labouré College
Essentials of Statistics
MAT 3410
ANSWERS
Chapter 7.2 through 7.3 Quiz Prep
Matthew Gitlin and colleagues followed a population of chronic dialysis patients to study
their patterns of getting blood transfusions and payments to Medicare.
The Charlson Co-morbidity Index (“Charlson”) was invented by Charlson in 1987. It
calculates a score for each person based on their co-morbidities (diabetes, congestive
heart failure) so that patients can be fairly compared as to how “sick” they are; a high
score means high co-morbidity, and a low score means low co-morbidity.
Gitlin et al. found that, among their patients, Charlson was normally distributed, with:


μ = 4.3
ϭ = 2.5
Answer the questions below based on the parameters listed above from the Gitlin
article.
Reference: Gitlin, M., Lee, J. A., Spiegel, D. M., Carson, J. L., Song, X., Custer, B. S., ... & Ashfaq, A.
(2012). Outpatient red blood cell transfusion payments among patients on chronic dialysis. BMC
nephrology, 13(1), 145.
The following questions do not require the use of the z table. Round to 2-digits after the
decimal unless otherwise specified in the question.
1.
What is the z-score corresponding to a Charlson score of 5? 0.28
(1 point) (round to the nearest hundredth) (5-4.3)/2.5 = 0.28. It is positive
because 5 is bigger than 4.3.
2.
What is the z-score corresponding to a Charlson score of 2? -0.92
(1 point) (round to the nearest hundredth) (2-4.3)/2.5 = -0.92. It is negative
because 2 is smaller than 4.3.
3.
What are the z scores corresponding to a range of Charlson score of 1.5 to 4.5?
-1.12 to 0.08
(2 points) (round to the nearest hundredth) (1.5-4.3)/2.5 = -1.12, and (4.5-4.3)/2.5
= 0.08.
4.
What Charlson score corresponds with a z score of 2.2? Round to 1-digit past
the decimal.
9.8
(1 point) (round to the nearest tenth) (2.2 * 2.5) + 4.3 = 9.8 (use formula).
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5.
What Charlson score corresponds with a z score of -1.1? Round to 1-digit past
the decimal
1.6
(1 point) (round to the nearest hundredth) (-1.1 * 2.5) + 4.3 = 1.6.
6.
You learn a person in the population had a Charlson of 5. Would you consider
that unusual? (2 points)
a.
b.
c.
d.
7.
No, it is within 1 SD of the mean
Yes, it is within 2 SD’s of the mean
Yes, it is more than 2 SD’s higher than the mean
Yes, it is more than 2 SD’s lower than the mean
For a random person in the population studied, what is the probability the person
will have a Charlson more than 2?
2
4.3
Record probability here:
0.8212
(2 - 4.3)/2.5 = -0.92, in z table, probability = 0.1788. This is the small white piece
because it is <0.5000 (50%), so we take 1 - 0.1788 to get the big red piece which
is 0.8212 or 82%.
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8.
Which of the following shapes corresponds to the section of the z distribution
associated with a Charlson of more than 2?
D IS THE CORRECT ANSWER
9.
For a random person in the population studied, what is the probability the person
will have a Charlson less than 4.5?
4.3 4.5
Record probability here:
0.5319
(1 point.) (4.5 - 4.3)/2.5=0.08. Look up 0.08 in z-table and find 0.5319. You are
looking for the big piece, and this is >0.5000 (50%) so keep it.
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©Copyright Monika Maya Wahi 2015
10.
For a random person in the population studied, what is the probability the person
will have a Charlson greater than 1.5 but less than 5?
1.5
4.3
5
Record probability here:
0.4789
(2 points.) (1.5 - 4.3)/2.5 = -1.12. Look that up in z-table, and find probability =
0.1314. This is the little piece at the bottom that is white (about 13%).
Next, (5 - 4.3)/2.5 = 0.28. Look that up in z-table, and find probability = 0.6103.
That is the whole shape below 5 including the little white piece at the bottom.
This is why it is big (61%).
Now we subtract the little white piece off from the big piece we got from the
second step. 0.6103 – 0.1314 = 0.4789 (48%), which is the middle red piece.
11.
Which of the following shapes corresponds to the section of the z distribution
associated with a Charlson of greater than 1.5 but less than 5?
C IS THE CORRECT ANSWER
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©Copyright Monika Maya Wahi 2015
12.
Imagine that the investigators became concerned at some of the high Charlson
scores they saw. They decided to give free treatment to people with Charlson
scores of 7 or higher. What percentage of patients will receive free treatment
from the investigators?
4.3 7.0
Record percentage here:
14%
(1 point) (7 - 4.3)/2.5 = 1.08. Look this up in z-table, and find probability is
0.8599. Since 0.8599 is >0.5000 (50%), you know you got the big piece, which is
the white piece below 7. Since you want the red piece at the top, take 1 – 0.8599
= 0.1401, or 14%.
13.
Imagine that the investigators realized that they cannot afford to treat more than
3% of their population. Given this, at what Charlson score will patients receive
free treatment? Round Charlson score to 1-digit past the decimal.
4.3
9.0
Record Charlson here:
9.0
(1 point) When you are given a % in the question, dig around in the middle of the
z-table and look for a probability closest to the % given. Here, 3% is the same as
0.03, so we look for the in the middle of the z table and we find the closest to this
at 0.0301, which is at z score -1.88. Now we can use that z score in the formula:
Because the z score you found is negative, you know it is for the bottom 3%,
because that is below the mean (-1.88). The top 3% is the positive version of
this, which is 1.88. Therefore, use the positive one when finding the x for a “top”
%, and the negative one for finding an x for the “bottom” %.
We are looking for the “top’ %, so we will use the positive version to calculate x:
(1.88)(2.5) + 4.3 = 9.0.
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