Download z-score homework - rivier.instructure.com.

Z-score Assignment
Standard Distribution
Normal Curve Standard Deviation : 1) Review the percentage values under the normal
curve at the end of your Normal Curve & Z-scores Class Notes). 2) Watch for your
positives & negatives in both the formulas & proportion problems. 3) If the question asks
for percentages, be sure to display your answer in percentages. 4) The Unit Normal Table
in the back of the text limits you to 2 spaces to the right of the decimal for z-scores (so be
sure to round correctly) & 4 spaces to the right of the decimal for proportions under the
normal curve. For all other answers, you will want to round as far out as you can. I usually
work at least up to 5 or 6 spaces to the right of the decimal as the farther to the right of the
decimal you round, the more accurate your outcome will be.
There are examples displayed for you for each set of problems. Formulas & additional
commentary is also available to assist you in working each problem correctly.
Z-scores
1. A distribution has a standard deviation of σ = 10. For each of the following z-scores,
determine whether the location is above or below the mean & determine how many
points away from the mean. For example, z = +1.00 corresponds to a location that is
above the mean by 10 points.
Example: Z = +3.00 = 30 points above the mean (1 z-score is = 10 points)
a. Z = +2.00
b. Z = +.50
c. Z = -2.00
d. Z = -1.50
2. For a population with µ = 45 & σ = 7, find the z-score for each of the following X
values (Use your z-score conversion formula in your class notes: z = x - µ / σ).
Example: X = 22 : 22-45/7 = - 23/7 = - 3.2857
a. X = 47
b. X = 35
c. X = 40
d. X = 60
e. X = 55
f. X = 42
3. A population has a mean of µ = 70 & a standard deviation of σ = 8. Find the z-score
corresponding to each of the following scores. (Use your z-score conversion formula
in your class notes: z = x - µ / σ).
Example: X = 72 : 72-70/8 = 2/8 = +.25
a. X = 74
b. X = 68
c. X = 86
d. X = 55
e. X = 70
f. X = 75
4. A sample has a mean of M = 75 & a standard deviation of s = 10. Find the X value
corresponding to each of the following z-scores for this sample (look at your
formulas for converting values back from a z-score to a raw score or X value. Watch
out for your positive & negative values: X = µ + zσ. Or as a sample: X = M + zs) The
difference being the ‘symbols’ only, but the calculation process is the same.
Example: Z = 2.00 : 75 + (2.00)(10) = 75 + 20 = 95
a. Z = 1.50
b. Z = -2.30
c. Z = -0.80
d. Z = 0.40
e. Z = -1.20
f. Z = 2.10
5. For this set of problems, you will be adding up the distance between 2 z-scores based
upon the area covered under the normal curve:
For the set of problems below, use the normal curves copied again for you just above. Mark
the two points (z-scores) noted in each problem on the normal curve & then add the
percentages identified in the space between these two z-score points. This will then tell you
the percentage of the area covered under the normal curve between those two points.
Example: What is the distance under the normal curve between a z-score of -1.5 & a zscore of +.05 : 9.2 + 15 +19.1 +19.1 = 62.40% (if I color in the area under the normal curve
between the two z-scores, - 1.5 & +.05, the percentages added together in this example
equals the percentage of space covered under the normal curve between these two z-scores)
a. What is the distance under the normal curve between a Z-score of 0 & a Z-score of
2?
b. What is the distance under the normal curve between a Z-score of 0 & a Z-score of
- 2?
c. What is the distance under the normal curve between a Z-score of -1 & a Z-score of
1?
d. What is the distance under the normal curve between a Z-score of 2.5 & a Z-score of
3?
e. What is the distance under the normal curve between a Z-score of -0.5 & a Z-score
of 2?
f. What is the distance under the normal curve between a Z-score of -2.5 & a Z-score
of -1?
g. What is the distance under the normal curve between a Z-score of 0 & a Z-score of
3?
h. What is the distance under the normal curve between a Z-score of 1 & a Z-score of
2.5?
6. On Tuesday afternoon, Bill earned a score of X = 73 on an English test with µ = 65
& σ = 8. The same day, John earned a score of X = 63 on a math test with µ = 57 & σ
= 3. Who should expect the better grade, bill or John? Explain your answer.
7. Go to the Unit Normal Table in the back of your text (Appendix B). Identify the
proportions associated with the z-scores below. First, indicate the 4 values to the
right of the decimal in your answer, & then convert these values to percentages by
moving the decimal over 2 spaces to the right & then adding the % symbol.
Remember to indicate all 4 numbers in your answers.
Example: Z-score of 2.40; proportion in the body : .9918 or 99.18%
a. Z-score of .49; proportion in the body:
b. Z-score of 1.22; proportion in the tail:
c. Z-score of .50; proportion between the mean & z-score:
d. Z-score of 3.07; proportion in the body:
e. Z-score of .19; proportion in the tail:
f. Z-score of 2.33; proportion between the mean & z-score:
Once raw scores are converted to standard scores, it is often that problems include
identifying the probability of obtaining a set of scores. Through the use of the Unit
Normal Table & given a particular z-score, probability statements can be made. For
#6, you are to identify the probabilities by identifying the correct value associated
with the z-score.
When determining the probability of obtaining a certain value, follow these steps:
(1) Draw a normal curve & mark on the curve where the value would fall (for
instance, a) states -1.00, so you would mark on the normal curve -1, or 1 SD
below the mean or the center of the curve). You don’t have to include the
drawing in this assignment, but it helps immensely in determining proportion as
a frame of reference.
(2) Go to the back of your text, Appendix B, the Unit Normal Table & find the zscore 1.00 under the z-score column (The Z-scores in the Unit Normal Table
does not utilize negative values since the other columns are used to determine
distance under the normal curve)
(3) The question will ask you to find the probability of obtaining a score greater
than, less than or b/t the mean & the z-score. If the question asks “greater than,”
then you are looking at the area to the right of the z-score. Color that area in
under the normal curve that you drew. Since the first problem says “greater
than,” that means that you will color in all the space from -1.00 to the right
under your normal curve. If the problem asks “less than,” then you are looking
to the area to the left of the z-score.
(4) You have 3 options in the Unit Normal Table to choose from. The first column
says “proportion in the body.” This means if you are looking at an area under
the normal curve of 50% or greater, this would represent the body of the curve.
The next column says “proportion in the tail.” If less than 50% of the area under
the normal curve is colored in, then you are looking at the area in the tail. Then
the last column, “area b/t the mean & the z-score” is just that. The area you
would be looking at between the z-score & the mean, or the very center of the
distribution. For our first problem, since all of the positive side of the normal
curve is colored in including 1 SD on the negative side, then more than 50% of
the area is colored in, so we would be looking at the proportion in the body.
(5) Write down the .xxxx value correlating with the z-score & the proportion you
are looking at. Then, move the decimal over 2 places the right. This will then tell
you the probability of obtaining a score that falls w/in the space you have
colored in. For the first problem, since we are looking at the area in the body,
the value corresponding to our z-score of 1.00 is .8413, then our probability
would be 84.13%
Example: What is the probability of obtaining a score below a z-score of +1.44 : .9251
or 92.51% (The z-score here in this example is positive, so I am working with a z-score
on the right side of the normal curve. Since the question states “below” or “less than,”
that means that I am looking at the area to the left of the z-score +1.44. Since I am
working with the left of this score, that means that I am including all of the space within
the negative side of the normal curve plus 1.44 standard deviations to the right of the
normal curve, which covers more than 50% of the distribution. Since I am working
with more than 50% of the distribution, then I am looking for the proportion in the
body, which is .9251 or 92.51%
8. What is the probability of obtaining a score ….
a. Greater than a z-score of – 1.00
b. Greater than a z-score of +2.00
c. Less than a z-score of + 1.25
d. Less than a z-score of - .86
e. Between a z-score of + .50 & the mean
f. Between a z-score of - .92 & the mean
9. Re: Your SPSS Standard Score assignment this week. Review the raw scores &
their corresponding z-scores in your final data screen. In the questions below,
identify the z-score in question & its corresponding raw score value.
Example: Identify the corresponding raw score that is second highest on the
distribution: The math raw score of 35 by participant #5. (Since the problem is
asking me to find the next to the ‘highest’ value, I know I can eliminate all
negative z-scores as all of those scores fall below the mean, so I want to look at
all of the positive scores. I can first find the z-score that is the highest & then
locate the z-score that is next to the highest value, which is the z-score for the
math test = .94729. The raw score that is associated with this z-score is the math
raw score of 35 from participant #5).
a) Identify the corresponding raw score that falls highest on the distribution:
b) Identify the corresponding raw score that falls lowest on the distribution:
c) Identify the corresponding raw score that falls closest to the mean on the
positive side of the distribution:
d) Identify the corresponding raw score that falls closest to the mean on the
negative side of the distribution: