Introduction to working with z-scores on standard and modified Normal curves.

1. Working with z-scores
z by Shamana ©

2. Professor Adams has 140 students who wrote a statistics test. If the marks are
approximately normally distributed:
• how many students should have a 'B' mark (i.e., 70 to 79 percent)?
• how many students should have failed (i.e., less than 50 percent)?
• how high must she set the mark for an 'A' if she wants 5 percent of the
students to get an A?
• how high must she set the passing mark if she wants only the top 75
percent of the marks to be passing marks?

3. A group of 60 Applied Math students obtained the following scores on the
final examination last year.
60 64 59 56 48 39 65 64 63 53
92 64 69 72 77 51 75 53 76 79
90 48 39 76 46 27 23 55 65 82
76 54 72 27 72 48 67 40 85 36
96 68 72 81 36 55 51 94 69 57
68 92 39 48 37 63 82 43 72 37
Draw a suitable histogram that has 10 bars.

4. Working with the Normal Distribution
Example: The graph at right represents
the marks of a large number of students
where the mean mark is 69.3 percent
and the standard deviation is 7 percent,
and the distribution of marks is
approximately normal.
62.3% 69.3% 76.3%

5. Working with the Normal Distribution
Example: The graph at right represents
the marks of a large number of students
where the mean mark is 69.3 percent
and the standard deviation is 7 percent,
and the distribution of marks is
approximately normal.
We know that if the marks are approximately normally distributed, then approximately:
• 68 percent of the marks are between 62.3 percent and 76.3 percent (i.e., µ ± 1σ)
• 34% of the marks are between 69.3 percent and 76.3 percent (i.e., between µ and µ + 1σ)
• 50% of the marks are below 69.3 percent (i.e., less than µ)
• 16% of the marks are above 76.3 percent (i.e., greater than µ + 1σ)

6. RECALL
North American women have a mean height of 161.5 cm and a
standard deviation of 6.3 cm.
(a) A car designer designs car seats to fit women taller than 159.0 cm.
What is the z-score of a woman who is 159.0 cm tall?
(b) The manufacturer designs the seats to fit women with a maximum
z-score of 2.8. How tall is a woman with a z-score of 2.8?

7. ShadeNorm(Lo z, Hi z) DICTIONARY
[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

8. Case 1(a): Calculate the Percentage of Scores Between Two Given 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 'B'
mark (i.e., 70 percent to 79 percent)? Assume that the marks are
normally distributed.