Download Lesson 7.5 Areas Under Any Normal Curve – Checking for

Lesson 7.5 Areas Under Any Normal Curve – Checking for
Normality Notes
Statistics
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Standard Normal Distribution
 Normal distribution with mean µ = 0 and standard deviation σ = 1
 z Score Formula: z 
x

, where x is the raw data, µ is the mean and σ is
the standard deviation.
How to work with Normal Distributions
 Sometimes the raw score, x, is given. When trying to find probability for a
random variable that follows the normal distribution pattern, you must first
convert the raw score, x, to a z value.
x
 Use z 
, where x is the raw data, µ is the mean and σ is the standard

deviation to change the x to the z value .
Example 1: Suppose x has a normal distribution with µ= -1 and σ = 3.
a. Convert the statement x ≤ 2 to a statement about z.
b. What is the area under the standard normal curve to the left of z = 1.5?
c. What is the probability P(z ≤ 1.5)?
d. What is the probability P(z ≤2)?
Lesson 7.5 Areas Under Any Normal Curve – Checking for
Normality Notes
Statistics
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Example 2: Find the z value such that 3% of the area under the standard normal
curve lies to the left of z.
a. Draw a sketch of the standard normal distribution showing the area.
b. Find the area to the left of z.
c. Find the corresponding z.
d. Suppose the time to complete a test is normally distributed with µ=40
minutes and σ=5 minutes. After how many minutes can we expect all but
about 3% of the tests to be completed?
e. Find a z value such that 3% of the area under the standard normal curve lies
to the left of z.
f. Compare the z value in part (c) with the z value in part (e). Is there any
relationship between the z values?
Lesson 7.5 Areas Under Any Normal Curve – Checking for
Normality Notes
Statistics
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Checking for Normality
 How to determine whether data have a normal distribution…
1. Histogram: Make a histogram. Histogram should be symmetrical or bell
shaped.
2. Outliers: For normal distribution, there should be no more than one
outlier. Use a box and whiskers plot. Find the outliers by
3. Skewness: Normal distributions are symmetrical. One measure of
skewness for sample data is given by Pearson’s index.
3  X  median 
Pearson’s index =
. An index value greater than 1 or
s
less than -1 indicates skewness. Skewed distributions are not normal.
4. Normal quantile plot (or normal probability plot): Plot the data with the
help of your calculator. If the points lie close to a straight line, the data
come from a distribution that is approximately normal.
a. Enter data in table.
b. Press STATPLOT and select one of the plots.
c. Highlight ON.
d. Then highlight the 6th plot option.
e. Choose the correct list.
f. Choose Y as the data axis.
g. Press GRAPH.
h. Adjust window is necessary.
i. Analyze the graph to determine if it is linear.
Lesson 7.5 Areas Under Any Normal Curve – Checking for
Normality Notes
Statistics
Example 3: Consider the following data.
19
19
19
16
21
14
23
17
19
20
20
13
16
17
19
18
19
17
21
24
19
21
22
20
20
20
24
17
20
22
21
18
20
22
16
15
21
23
21
18
15
25
a. Make a histogram and box and whiskers plot for this data.
b. Use Pearson’s index to check for skewness.
c. Make a normal quantile plot for this data.
d. Interpret the results.
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18
18
19
18
24
23
22
20
Lesson 7.5 Areas Under Any Normal Curve – Checking for
Normality Notes
Statistics
Example 4: Consider the following data.
5
2
3
4
3
3
4
4
3
2
a. Make a histogram and box and whiskers plot for this data.
b. Use Pearson’s index to check for skewness.
c. Make a normal quantile plot for this data.
d. Interpret the results.
Assignment: p. 270 #3, 4, 8, 12, 20, 28, 29, 30
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