Download Lesson 1.1.2

Introduction
Previous lessons demonstrated the use of the standard
normal distribution. While distributions with a mean of 0
and a standard deviation of 1 are rare in the real world,
there is a formula that allows us to use the properties of
a standard normal distribution for any normally
distributed data. With this formula, we can generate a
number called a z-score to use with our data. This
makes the normal distribution a powerful tool for
analyzing a wide variety of situations in business and
industry as well as the physical and social sciences.
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1.1.2: Standard Normal Calculations
Introduction, continued
Using and understanding z-scores requires a deeper
understanding of standard deviation. In the previous
sub-lesson, we found the standard deviations of small
data sets. In this lesson, we will explore how to use
z-scores and graphing calculators to evaluate large
data sets.
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1.1.2: Standard Normal Calculations
Key Concepts
• Recall that a population is all of the people or things of
interest in a given study, and that a sample is a subset
(or smaller portion) of the population.
• Samples are used when it is impractical or inefficient
to measure an entire population. Sample statistics are
often used to estimate measures of the population
(parameters).
• The mean of a sample is the sum of the data points in
the sample divided by the number of data points, and
is denoted by the Greek letter mu, μ.
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1.1.2: Standard Normal Calculations
Key Concepts, continued
• The mean is given by the formula
,
where each x-value is a data point and n is the total
number of data points in the set.
• From a visual perspective, the mean is the balancing
point of a distribution.
• The mean of a symmetric distribution is also the
median of the distribution.
• The median is the middle value in a list of numbers.
• Both the mean and median are at the center of a
symmetric distribution.
1.1.2: Standard Normal Calculations
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Key Concepts, continued
• The standard deviation of a distribution is a measure of
variation.
• Another way to think of standard deviation is “average
distance from the mean.” The formula for the standard
n
deviation is given by s =
2
(x
m
)
å i
i=1
, where s (the
n
lowercase Greek letter sigma) represents the standard
n
deviation, xi is a data point, and
sum from 1 to n data points.
å means to take the
i =1
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1.1.2: Standard Normal Calculations
Key Concepts, continued
• Summation notation is used in the formula for
calculating standard deviation; it is a symbolic way to
represent the sum of a sequence.
• Summation notation uses the uppercase version of the
Greek letter sigma, Σ.
• After calculating the standard deviation, σ, you can use
this value to calculate a z-score.
• A z-score measures the number of standard deviations
that a given score lies above or below the mean. For
example, if a value is three standard deviations above
the mean, its z-score is 3.
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1.1.2: Standard Normal Calculations
Key Concepts, continued
• A positive z-score corresponds to an individual score that
lies above the mean, while a negative z-score corresponds
to an individual score that lies below the mean.
• By using z-scores, probabilities associated with the
standard normal distribution (mean = 0, standard
deviation = 1) can be used for any non-standard normal
distribution (mean ≠ 0, standard deviation ≠ 1).
• The formula for calculating the z-score is given by
x-m
, where z is the z-score, x is the data point, μ is
z=
s
the mean, and σ is the standard deviation.
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1.1.2: Standard Normal Calculations
Key Concepts, continued
• z-scores can be looked up in a table to determine the
associated area or probability.
• The numerical value of a z-score can be rounded to the
nearest hundredth.
• Graphing calculators can greatly simplify the process of
finding statistics and probabilities associated with
normal distributions.
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1.1.2: Standard Normal Calculations
Common Errors/Misconceptions
• calculating and applying a z-score to a distribution that
is not normally distributed
• using the area to the left of the z-score when the area
to the right of the z-score is the area of interest and
vice versa
• misreading the table with the associated probability
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1.1.2: Standard Normal Calculations
Guided Practice
Example 1
In the 2012 Olympics, the mean finishing time for the
men’s 100-meter dash finals was 10.10 seconds and the
standard deviation was 0.72 second. Usain Bolt won the
gold medal, with a time of 9.63 seconds. Assume a
normal distribution. What was Usain Bolt’s z-score?
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1.1.2: Standard Normal Calculations
Guided Practice: Example 1, continued
1. Write the known information about the
distribution.
Let x represent Usain Bolt’s time in seconds.
μ = 10.10
s = 0.72
x = 9.63
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1.1.2: Standard Normal Calculations
Guided Practice: Example 1, continued
2. Substitute these values into the formula
for calculating z-scores.
The z-score formula is z =
z=
x-m
s
=
9.63 - 10.10
0.72
x-m
s
.
» -0.65
Usain Bolt’s z-score for the race was –0.65.
Therefore, his time was 0.65 standard deviations
below the mean.
✔
1.1.2: Standard Normal Calculations
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Guided Practice: Example 1, continued
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1.1.2: Standard Normal Calculations
Guided Practice
Example 2
What percent of the values in a normal distribution are
more than 1.2 standard deviations above the mean?
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1.1.2: Standard Normal Calculations
Guided Practice: Example 2, continued
1. Sketch a normal curve and shade the
area that corresponds to the given
information.
Start by drawing a number
line. Be sure to include the
range of values –3 to 3.
Create a vertical line
at 1.2. Shade the region
to the right of 1.2.
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1.1.2: Standard Normal Calculations
Guided Practice: Example 2, continued
2. Use a table of z-scores or a graphing
calculator to determine the shaded area.
A z-score table can be used to determine the area.
Since the area of interest is 1.2 standard deviations
above the mean and greater, we need to look up the
area associated with a z-score of 1.2.
The following table contains z-scores for values
around 1.2σ.
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1.1.2: Standard Normal Calculations
Guided Practice: Example 2, continued
z
0.0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549
0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852
0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133
0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.834 0.8365 0.8389
1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621
1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830
1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015
1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177
1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319
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1.1.2: Standard Normal Calculations
Guided Practice: Example 2, continued
To find the area to the left of 1.2, locate 1.2 in the
left-hand column of the z-score table, then locate the
remaining digit 0 as 0.00 in the top row. The entry
opposite 1.2 and under 0.00 is 0.8849; therefore, the
area to the left of a z-score of 1.2 is 0.8849 or 88.49%.
We are interested in the area to the right of the zscore. Therefore, subtract the area found in the table
from the total area under the normal distribution, 1.
1 – 0.8849 = 0.1151
The area greater than 1.2 standard deviations under
the normal curve is about 0.1151 or 11.51%.
1.1.2: Standard Normal Calculations
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Guided Practice: Example 2, continued
Alternately, you can use a graphing calculator to
determine the area of the shaded region.
Note: The lower bound is 1.2, but the upper bound is
infinity, so any large positive integer will work as the
upper bound value. Use 100 as the upper bound.
Since this problem is based on standard deviations
under the standard normal distribution, the mean = 0
and the standard deviation = 1.
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1.1.2: Standard Normal Calculations
Guided Practice: Example 2, continued
On a TI-83/84:
Step 1: Press [2ND][VARS] to bring up the
distribution menu.
Step 2: Arrow down to 2: normalcdf. Press [ENTER].
Step 3: Enter the following values for the lower
bound, upper bound, mean (μ), and standard
deviation (σ). Press [,] after typing each
value. Lower: [1.2]; upper: [100]; μ: [0]; σ: [1].
Step 4: Press [ENTER] to calculate the area of the
shaded region.
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1.1.2: Standard Normal Calculations
Guided Practice: Example 2, continued
On a TI-Nspire:
Step 1: Press the [home] key.
Step 2: Arrow over to the spreadsheet icon and
press [enter].
Step 3: Press the [menu] key. Arrow down to 4:
Statistics, then arrow right to bring up the
sub-menu. Arrow down to 2: Distributions and
press [enter].
Step 4: Arrow down to 2: Normal Cdf. Press [enter].
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1.1.2: Standard Normal Calculations
Guided Practice: Example 2, continued
Step 5: Enter the values for the lower bound, upper
bound, mean (μ), and standard deviation
(σ), using the [tab] key to navigate between
fields. Lower Bound: [1.2]; Upper Bound:
[100]; μ; [0]; σ: [1]. Tab down to “OK” and
press [enter].
Step 6: The values entered will appear in the
spreadsheet. Press [enter] again to
calculate the area of the shaded region.
The area returned on either calculator is
about 0.1151 or 11.51%.
1.1.2: Standard Normal Calculations
✔
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Guided Practice: Example 2, continued
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1.1.2: Standard Normal Calculations