Working with z-scores: the three "meanings" of the area under the Normal Curve.

1. Working the
scores
The letter Z by flickr user Admit One

2. ShadeNorm(Lo z, Hi z)
[shades area under std. normal curve]
σ
ShadeNorm(Lo value, Hi value, mean, std. dev.)
[shades area under modified normal curve]
µ-5σ
σ
µ+5σ

3. Properties of a Normal Distribution
The 68-95-99 Rule
Generally speaking, approximately:
• 68% of all the data in a normal distribution lie within the 1
standard deviation of the mean,
• 95% of all the data lie within 2 standard deviations of the
mean, and
• 99.7% of all the data lie within 3 standard deviations of the
mean.

4. Case 1(a): Calculate the Percentage of Scores Between Two Given
Scores
HOMEWORK
The mean mark for a large number of students is 69.3 percent
with a standard deviation of 7 percent. What percent of the
students have a 'B' mark (i.e., 70 percent to 79 percent)?
Assume that the marks are normally distributed.

5. HOMEWORK
Find the percent of z-scores in a standard normal distribution that are:
(b) above z = -2.35
(a) below z = 0.52
(d) between z = 0.55 and z = 0.15
(c) between z = -1.11 and z = 0.92

6. HOMEWORK
Find the z-score if the area under a standard normal curve:
(a) to the left of z is 0.812
(b) to the right of z is 0.305
(c)to the right of z is 0.785

7. HOMEWORK
Determine the values for z and x.

8. The 3 meanings of a Shaded Normal Curve
area
zlow zhigh
The shaded area under a normal curve between two
z-scores is interpreted, simultaneously, as:
• an area
(the area under the normal curve)

9. The 3 meanings of a Shaded Normal Curve
%
zlow zhigh
The shaded area under a normal curve between two
z-scores is interpreted, simultaneously, as:
• an area
(the area under the normal curve)
• a percentage
(the percentage of all values in a data set that lie between two
particular z-scores)

10. The 3 meanings of a Shaded Normal Curve
P(E)
zlow zhigh
The shaded area under a normal curve between two
z-scores is interpreted, simultaneously, as:
• an area
(the area under the normal curve)
• a percentage
(the percentage of all values in a data set that lie between two
particular z-scores)
• a probability
(the probability that a particular z-score falls between two given
z-scores)

11. The 3 meanings of a Shaded Normal Curve
area
%
P(E)
zlow zhigh
The shaded area under a normal curve between two
z-scores is interpreted, simultaneously, as:
• an area
(the area under the normal curve)
• a percentage
(the percentage of all values in a data set that lie between two
particular z-scores)
• a probability
(the probability that a particular z-score falls between two given
z-scores)

12. Case 1(c): Calculate the Number of Scores
1200 light bulbs were tested for the number of hours of life. The mean
life was 640 hours with a standard deviation of 50 hours. Assume that
quot;life in hoursquot; of light bulbs is normally distributed.
(a) What percent of light bulbs lasted between 600 and 700 hours?
(b) How many light bulbs should be expected to last between
600 and 700 hours?

13. Case 1(b): Calculate the Percentage of Passing Scores
The mean mark for a large number of students is 69.3 percent with a
standard deviation of 7 percent. What percent of the students have a
passing mark if they must get 60 percent or better to pass? Assume that
the marks are normally distributed.
HOMEWORK

14. Case 2(a): Calculate the Percentage of Scores Between
Two Z-Scores HOMEWORK
(Case 2) If we know two z-scores of a standard normal distribution,
we can find the percentage of scores that lie between them. The
procedure is similar to that used in the previous examples.
Sample question(s):What percent of scores lie between z =
0.87 and z = 2.57?
OR
What is the probability that a score will fall between z = 0.87
and z = 2.57?
OR
Find the area between z = 0.87 and z = 2.57 in a standard
normal distribution.

15. Case 2(b): Calculate the Percentage of Scores Between Two
Z-Scores HOMEWORK
Find the probability of getting a z-score less than 0.75 in a standard
normal distribution.

16. Case 3(b): Find the Z-Value that Corresponds to a Given Probability
HOMEWORK
What is the z-score if the probability of getting less than this z-score is
0.750?

17. Case 3(a): Find the Z-Score that Corresponds to a Given Probability
HOMEWORK
If we know the probability of an event, we can find the z-score that
corresponds to this probability. This is the reverse of what we did in
Case 2.
Sample question:
What is the z-score if the probability of getting more than this
z-score is 0.350?