Download AL7-10

5-Minute Check on Activity 7-9
1. What population parameter is a measure of spread?
Standard deviation, σ
2. What measure of spread is resistant?
Inter-quartile range, IQR
3. Given the following data: IQR = 10, Q1 = 20 and Q3 = 40; determine
the upper and lower fences and if 6 and 56 are outliers
UF = Q3 + 1.5×IQR = 40 + 1.5×10 = 55 since 56>55, its an outlier
LF = Q1 - 1.5×IQR = 20 - 1.5×10 = 5 since 6>5, its not an outlier
4. What 5 numbers are in the 5-number summary?
Min, Q1, Q2 (median), Q3 and Max
5. Label the box-plot below:
*
Min
Q1
Q2
Q3
Click the mouse button or press the Space Bar to display the answers.
Max - outlier
Activity 7 - 10
What Is Normal?
Objectives
• Identify a normal distribution
• List the properties of a normal curve
• Determine the z-score of a given numerical data
value in a normal distribution
• Identify the properties of a standard normal curve
• Solve problems using z-scores of a standardized
normal curve
Vocabulary
• Normal Distribution – an important distribution in the study of
probability and statistics
• Normal Curve – is a bell-shaped curve with specific properties
• z-scores – a standardized score based on how many standard
deviations a data point is away from the mean
Activity
The following collection of data gives the heights, in
inches, of 35 randomly selected 11th grade male students.
The measurements were made to the nearest inch. Type
them into L1 in your calculator and it can help you with
parts b and d.
63
64
65
65
66
66
66
67
67
67
67
67
68
68
68
68
68
68
69
69
69
69
69
70
70
70
70
71
71
72
72
73
74
75
76
a) Complete the following frequency distribution table
Number
1
1
2
3
5
6
5
4
2
2
1
1
1
1
Height
63
64
65
66
67
68
69
70
71
72
73
74
75
76
Activity
Number
1
1
2
3
5
6
5
4
2
2
1
1
1
1
Height
63
64
65
66
67
68
69
70
71
72
73
74
75
76
b) Construct a histogram from the frequency distribution
in part a
63
64
65
66
67
68
69
70
71
72
73
74
75
76
c) Does the distribution appear to be skewed or
symmetric? Explain
Symmetric with a little right skewness (towards tail)
Activity cont
The following collection of data gives the heights, in
inches, of 35 randomly selected 11th grade male students.
The measurements were made to the nearest inch.
63
64
65
65
66
66
66
67
67
67
67
67
68
68
68
68
68
68
69
69
69
69
69
70
70
70
70
71
71
72
72
73
74
75
76
d) Determine the mean, median and mode of the given
data set. Do your results verify your conclusion in
part c? Mean: 68.8 Median: 68 Mode: 68
Yes, symmetric  mean, median and mode about the same.
Right skewness pulls mean towards it (mean > median)
Properties of the Normal Density Curve
• The curve is bell-shaped with the highest point at μ, its
mean
• It is symmetric about a vertical line x = μ, its mean
• The mean, median, and mode are all equal
• 50% of the data values lie to the right of mean μ; and 50%
of the data values lie to the left of the mean μ
• The normal curve approaches the horizontal axis, but
never touches or crosses the axis
• The Empirical Rule applies:
– 68% of the data lies within 1 standard deviation of the mean
– 95% of the data lies within 2 standard deviations of the mean
– 99.7% of the data lies within 3 standard deviations of the mean
Normal Curves
• Two normal curves with different means (but
the same standard deviation) [on left]
– The curves are shifted left and right
• Two normal curves with different standard
deviations (but the same mean) [on right]
– The curves are shifted up and down
Empirical Rule
μ ± 3σ
μ ± 2σ
μ±σ
99.7%
95%
68%
2.35%
2.35%
34%
0.15%
34%
13.5%
μ - 3σ
μ - 2σ
μ-σ
13.5%
μ
μ+σ
μ + 2σ
0.15%
μ + 3σ
• Also known in statistics as the 68-95-99.7 Rule
Activity cont
Number
1
1
2
3
5
6
5
4
2
2
1
1
1
1
Height
63
64
65
66
67
68
69
70
71
72
73
74
75
76
e) Is this distribution Normal?
63
64
65
66
67
68
69
70
71
72
73
74
75
76
Approximately normal.
With slight right-skewness is not exactly normal.
Normal Probability Density Function
• There is a y = f(x) style function that
describes the normal curve:
1
-(x – μ)2
y = -------- e 2σ2
√2π
where μ is the mean and σ is the standard
deviation of the random variable x
• Only used in college post-calculus statistics
Standardized Score
• Z-score: The number of standard deviations a value
is away from its mean
x-μ
z = -------
where x is the data value
μ is the mean of the data set
 is the standard deviation of the data
• Values below the mean will be negative and values
above the mean will be positive
• Normal distributions can be written as N(,)
Example 1
A random variable x is normally distributed with μ=10
and σ=3.
a. Compute Z for x1 = 8
8 – 10
-2
Z = ---------- = ----- = -0.67
3
3
b. Compute Z for x2 = 12
12 – 10
2
Z = ----------- = ----- = 0.67
3
3
c. Compute Z for x3 = 10
10 – 10
0
Z = ----------- = ----- = 0.00
3
3
Example 2
The heights of 16-year old males are normally distributed
as N(68,2)
a. Compute Z for 71 inches
b. Compute Z for 64 inches
71 – 68
3
Z = ---------- = ----- = 1.50
2
2
64 – 68
4
Z = ----------- = ----- = -2.00
2
2
c. Compute Z for 62 inches
62 – 68
-6
Z = ----------- = ----- = -3.00
2
2
Example 3
A random variable x is normally distributed with μ=10
and σ=3. Given the z-scores, find the original value.
a. Find x for z = 2
x – 10
2 = ---------3
6 = x – 10
16 = x
b. Find x for z = -1
x – 10
-1 = ----------3
-3 = x – 10
7=x
c. Find x for z = -0.5
x – 10
-0.5 = ----------3
-1.5 = x – 10
8.5 = x
Example 4
What do the following z-scores mean?
a) z = -1
data point is one standard deviation below the mean
b) z = 2
data point is two standard deviations above the mean
c) z = 0
data point is at the mean
Summary and Homework
• Summary
– Normal curves are bell-shaped and symmetric about
its mean (which is equal to mode and median)
• Mean, , has 50% of data above it and below it
• Empirical Rule: 68-95-99.7% of the data lies within plus or
minus 1-2-3 standard deviations of the mean
– Standard normal curve has  = 0 and  = 1
– Z-scores are a measure of standard deviations
• Base on standard normal curve
• Positive values above mean; negative values below mean
• Homework
– pg 871 – 873; problems 1- 5