Introduction to working with the Standard Normal Curve.

1. Working with z-scores
z by Shamana ©

2. The data below shows the ages in years of 30 trees in an area of
natural vegetation.
37 15 34 26 25 38 19 22 21 28
42 18 27 32 19 17 29 28 24 35
35 20 23 36 21 39 16 40 18 41
Determine whether the data approximate the normal distribution.
USING the 68 -95-97 RULE

3. HOMEWORK
The following are the number of steak dinners served on 50
consecutive Sundays at a restaurant.
41 52 46 42 46 36 44 68 58 44
49 48 48 65 52 50 45 72 45 43
47 49 57 44 48 49 45 47 48 43
45 56 61 54 51 47 42 53 44 45
58 55 43 63 38 42 43 46 49 47
Draw a suitable histogram that has five bars.

4. HOMEWORK
The frequency table shows the ages of all the students in Senior 4
Math at Newberry High. Find the mean, μ. Then calculate the
percent of students older than the mean age. How does this
compare to the percent of students older than the mean age if the
distribution were a normal distribution?
Based on this answer, does it seem that the students' ages
approximate a normal distribution?
Age of Student 15 16 17 18 19 20 21 22
# of Students 1 7 42 24 7 4 2 1

5. Now let's try a problem involving Grouped Data
A machine is used to fill bags with beans. The machine is set to add
10 kilograms of beans to each bag. The table shows the weights of
277 bags that were randomly selected.
wt in kg 9.5 9.6 9.7 9.8 9.9 10.0 10.1 10.2 10.3 10.4 10.5
# of bags 1 3 13 25 41 66 52 41 25 7 3
(a) Are the weights normally distributed? How do you know?
(b) Do you think that using the machine is acceptable and fair to
the customers? Explain your reasoning.

6. Properties of a Normal Distribution
The shape of any normal distribution curve is determined by:
• the mean (μ)
• the standard deviation (σ)
Changing the mean will
shift the graph horizontally.
Changing the standard
deviation will change the
shape of the curve,
making it narrow or wide.

7. Properties of a Normal Distribution
The data are continuous and distributed evenly around the mean,
and the graph created by the data is a bell-shaped curve, as
shown in the examples below.
These curves represent data sets that have the same mean, but
different standard deviations. Which one has a larger standard
deviation (σ)?
How can you tell?

8. The contents in the cans of several cases of soft drinks were
tested. The mean contents per can is 356 mL, and the standard
deviation is 1.5 mL.
(a) Two cans were randomly selected and tested. One can held
358 mL, and the other can 352 mL. Calculate the z-score of
each.
(b) Two other cans had z-scores of -3 and 1.85. How many mL
did each contain?

9. Curving The Marks
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?