2.
HOMEWORK
The table shows the lengths in millimetres of 52 arrowheads.
16 16 17 17 18 18 18 18 19 20 20 21 21
21 22 22 22 23 23 23 24 24 25 25 25 26
26 26 26 27 27 27 27 27 28 28 28 28 29
30 30 30 30 30 30 31 33 33 34 35 39 40
(a) Calculate the mean length (b) Determine the lengths of arrowheads
and the standard deviation. one standard deviation below and one
standard deviation above the mean.
(c) How many arrowheads (d) What percent of the arrowheads
are within one standard are within one standard deviation
deviation of the mean? of the mean length?
3.
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?
4.
HOMEWORK
Four hundred people were surveyed No. of Videos Returned No. of Persons
to find how many videos they had 1 28
rented during the last month. 2 102
Determine the mean and median of the 3 160
frequency distribution shown below, 4 70
and draw a probability distribution 5 25
histogram. Also, determine the mode 6 13
by inspecting the frequency 7 0
distribution and the histogram. 8 2
5.
HOMEWORK
The table shows the weights weight interval mean interval # of infants
(in pounds) of 125 newborn 3.5 to 4.5 4 4
infants. The first column 4.5 to 5.5 5 11
shows the weight interval, 5.5 to 6.5 6 19
the second column the 6.5 to 7.5 7 33
average weight within each 7.5 to 8.5 8 29
weight interval, and the third 8.5 to 9.5 9 17
column the number of 9.5 to 10.5 10 8
newborn infants at each 10.5 to 11.5 11 4
weight. Total 125
(a) Calculate the mean weight and standard deviation.
(b) Calculate the weight of an infant at one standard deviation below the
mean weight, and one standard deviation above the mean.
(c) Determine the number of infants whose weights are within one
standard deviation of the mean weight.
(d) What percent of the infants have weights that are within one
standard deviation of the mean weight?
6.
HOMEWORK
A survey was conducted at DMCI to determine the number of music CDs
each student owned. The results of the survey showed that the average
number of CDs per student was 73 with a standard deviation of 24. After the
scores were standardized, the people doing the survey discovered that DJ
Chunky had a z-score rating of 2.9. How many CDs does Chunky have?
7.
HOMEWORK
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?
8.
HOMEWORK
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?
9.
The Normal Distribution
A Normal Distribution is a frequency distribution that can be represented by
a symmetrical bell-shaped curve which shows that most of the data are
concentrated around the centre (i.e., mean) of the distribution. The mean,
median, and mode are all equal. Since the median is the same as the mean,
50 percent of the data are lower than the mean, and 50 percent are higher.
The frequency distribution showing light bulb life, for example, shows that
the mean is 970 hours, and the hours of life for all the bulbs are spread
uniformly about the mean.
10.
The Normal Distribution
The diagram above represents a normal distribution. In real life, the data
would never fit a normal distribution perfectly. There are, however, many
situations where data do approximate a normal distribution. Some examples
would include:
• the heights and weights of adult males in North America
• the times for athletes to run 5000 metres
• the speed of cars on a busy highway
• the weights of loonies produced at the Winnipeg Mint
Note that all the examples represent continuous data.
11.
Properties of a Normal Distribution
• 99.7% of all the data lies within approximately 3 standard deviations of the
mean.
• All normal distributions are symetrical about the mean.
• Each value of mean and standard deviation determines a different normal
distributions.
• The area under the curve always equals one.
• The x-axis is an asymptote for the curve.
Frequency
Scores
Interactivate Normal Distribution
12.
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 standard deviations of the mean.
13.
Properties of a Normal Distribution
The curve is symmetrical about the mean. Most of the data are
relatively close to the mean, and the number of data decrease as you get
farther from the mean.
14.
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.
15.
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?
16.
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
17.
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.
18.
The chart shows the sizes of pants sold in one week at Dan's Clothing Shop.
38 34 42 40 42 32 30 34
40 38 40 38 36 42 44 42
38 36 36 42 36 46 40 38
40 36 44 36 38 34 38 40
Determine whether the data approximate the normal distribution.
19.
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.
20.
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.
21.
HOMEWORK
The diagram shows a normal
distribution with a mean of 28
and a standard deviation of 4.
The values represent the number
of standard deviations above and
below the mean. Replace the
numbers with raw scores.
22.
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
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