1. 13-5 The Normal Distribution
Today…
Utilize Normal Distribution to determine probabilities and position of data points.

2. What is the probability a dart would land in the red shaded area?

3. 13.5 The Normal Distribution
Distributions, so far, have been “empirical” or based on observation.
Now, distributions will be “theoretical” meaning they are based on theoretical probabilities.
The Key to Inferential Statistics:
A knowledge of Theoretical Distributions enables us to identify when actual observations are inconsistent with stated assumptions.

4. Consider the probability
when tossing a coin

5. Below is the Probability Distribution for the # of heads when flipping a coin ten times in a row.
The area of the rectangle is equal to the probability of the corresponding number of heads. Note that the probabilities add up to 1. This is always the case, since something must happen, and we have listed all possible outcomes.

6. Note that this histogram is narrower than the preceding one
Note that this histogram is narrower than the preceding one. Thus, as the number of tosses grows, the probability of the percentage of heads being close to 50% also grows. Also, note that this histogram has an even more pronounced bell shape.
There is a famous result of probability theory known as the central limit theorem which says that as the number of independent trials grow, the probability distribution of any repeated experiment, such as tossing a fair coin, approaches a curve known as a normal or Gaussian distribution. Gaussian distributions occur frequently in many areas of science, including psychology, sociology, and economics.

7. Discrete and Continuous Random Variables
12.5 – The Normal Distribution
Discrete and Continuous Random Variables
Discrete random variable: A random variable that can take on only certain fixed values.
The number of even values of a single die.
The number of heads in three tosses of a fair coin.
Continuous random variable: A variable whose values are not restricted.
The diameter of a growing tree.
The height of third graders.

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The graph of coin flips is discrete with 10 or 20 possibilities. The resulting graph is 10 or 20 distinct rectangles
When finding the probability distribution for intelligence, the variable is continuous and will result in a smooth bell-shaped curve.
© 2008 Pearson Addison-Wesley. All rights reserved

9. Definition and Properties of a Normal Curve
12.5 – The Normal Distribution
Definition and Properties of a Normal Curve
A normal curve is a symmetric, bell-shaped curve.
Any random continuous variable whose graph has this characteristic shape is said to have a normal distribution.
On a normal curve the horizontal axis is labeled with the mean and the specific data values of the standard deviations.
If the horizontal axis is labeled using the number of standard deviations from the mean, rather than the specific data values, then the curve the standard normal curve.

10. 12.5 – The Normal Distribution
Sample Statistics
Normal Curve
Standard Normal Curve
– 2.8
– 1.4
1.4
2.8
– 2
– 1 1 2
5.5
0 or
5.5
The area under the curve along a certain interval is numerically equal to the probability that the random variable will have a value in the corresponding interval.
The total area under the SNC is always 1 square unit.

11. 12.5 – The Normal Distribution
Normal Curves B S A C
S is standard, with mean = 0, standard deviation = 1
A has mean < 0, standard deviation = 1
B has mean = 0, standard deviation < 1
C has mean > 0, standard deviation > 1

12. Properties of Normal Curves
12.5 – The Normal Distribution
Properties of Normal Curves
The graph of a normal curve is bell-shaped and symmetric about a vertical line through its center.
The mean, median, and mode of a normal curve are all equal and occur at the center of the distribution.
Empirical Rule: the approximate percentage of all data lying within 1, 2, and 3 standard deviations of the mean.
within 1 standard deviation
68%
within 2 standard deviations
95%
within 3 standard deviations.
99.7%

13. 12.5 – The Normal Distribution
Empirical Rule
68%
95%
99.7%

14. Example: Applying the Empirical Rule
12.5 – The Normal Distribution
Example: Applying the Empirical Rule
A sociology class of 280 students takes an exam. The distribution of their scores can be treated as normal. Find the number of scores falling within 2 standard deviations of the mean.
A total of 95% of all scores lie within 2 standard deviations of the mean.
(.95)(280) =
266 scores

15. 12.5 – The Normal Distribution
Normal Curve Areas
The area under the curve is presented as one of the following:
Percentage (of total items that lie in an interval),
Probability (of a randomly chosen item lying in an interval),
Area (under the normal curve along an interval).
In a normal curve and a standard normal curve, the total area under the curve is equal to 1.

16. A Table of Standard Normal Curve Areas
12.5 – The Normal Distribution
A Table of Standard Normal Curve Areas
To answer questions that involve regions other than 1, 2, or 3 standard deviations, a Table of Standard Normal Curve Areas is necessary.
The table shows the area under the curve for all values in a normal distribution that lie between the mean and z standard deviations from the mean.
The percentage of values within a certain range of z-scores, or the probability of a value occurring within that range are the more common uses of the table.
Because of the symmetry of the normal curve, the table can be used for values above the mean or below the mean.

17. Areas under the SNC and z scores
A table gives the fraction of all scores in a normal distribution that lie between the mean and z standard deviations from the mean.
Page 766 in the Text

18. Example: Applying the Normal Curve Table
12.5 – The Normal Distribution
Example: Applying the Normal Curve Table
Use the table to find the percent of all scores that lie between the mean and 1.5 standard deviations above the mean.
z = 1.5
z = 1.50
Find 1.50 in the z column.
The table entry is .4332
Therefore, 43.32% of all values lie between the mean and 1.5 standard deviations above the mean. or
There is a probability that a randomly selected value will lie between the mean and 1.5 standard deviations above the mean.

19. Example: Applying the Normal Curve Table
12.5 – The Normal Distribution
Example: Applying the Normal Curve Table
Use the table to find the percent of all scores that lie between the mean and standard deviations below the mean.
z = –2.62
z = – 2.62
Find 2.62 in the z column.
The table entry is
Therefore, 49.56% of all values lie between the mean and 2.62 standard deviations below the mean. or
There is a probability that a randomly selected value will lie between the mean and 2.62 standard deviations below the mean.

20. Example: Applying the Normal Curve Table
12.5 – The Normal Distribution
Example: Applying the Normal Curve Table
Find the percent of all scores that lie between the given z-scores.
z = –1.7
z = 2.55
z = – 1.7
The table entry is
z = 2.55
The table entry is =
0.95
Therefore, 95% of all values lie between – 1.7 and 2.55 standard deviations.

21. Example: Applying the Normal Curve Table
12.5 – The Normal Distribution
Example: Applying the Normal Curve Table
Find the probability that a randomly selected value will lie between the given z-scores.
z = 0.61
z = 2.63
z = 0.61
The table entry is
z = 2.63
The table entry is
– =
0.2666
There is a probability that a randomly selected value will lie between 0.61 and 2.63 standard deviations.

22. Example: Applying the Normal Curve Table to Inverse Normal Calculation
12.5 – The Normal Distribution
Example: Applying the Normal Curve Table to Inverse Normal Calculation
Find the probability that a randomly selected value will lie above the given z-score.
z = 2.14
z = 2.14
The table entry is
Half of the area under the curve is
– =
0.0162
There is a probability that a randomly selected value will lie 2.14 standard deviations.

23. Example: Applying the Normal Curve Table
12.5 – The Normal Distribution
Example: Applying the Normal Curve Table
The volumes of soda in bottles from a small company are distributed normally with a mean of 12 ounces and a standard deviation .15 ounces. If 1 bottle is randomly selected, what is the probability that it will have more than ounces?
z = 2.2
The table entry is
12.33
Half of the area under the curve is
– =
0.0139
There is a probability that a randomly selected bottle will contain more than ounces.

24. Example: Finding z-scores for Inverse Normal Calculation
12.5 – The Normal Distribution
Example: Finding z-scores for Inverse Normal Calculation
Assuming a normal distribution, find the z-score meeting the condition that 39% of the area is to the right of z.
50% of the area lies to the right of the mean.
11%
= 0.11
39%
= 0.39
The areas from the Normal Curve Table are based on the area between the mean and the z-score.
area between the mean and the z-score =
0.50 – 0.39 =
0.11
From the table, find the area of or the closest value and read the z-score.
z-score = 0.28

25. Example: Finding z-scores for Given Areas
12.5 – The Normal Distribution
Example: Finding z-scores for Given Areas
Assuming a normal distribution, find the z-score meeting the condition that 76% of the area is to the left of z.
50% of the area lies to the left of the mean.
26%
= 0.26
50%
The areas from the Normal Curve Table are based on the area between the mean and the z-score.
0.5000
area between the mean and the z-score =
0.76 – 0.50 =
0.26
From the table, find the area of or the closest value and read the z-score.
z-score = 0.71

26. Expected Value
The expected value is found by multiplying the number in the sample by the probability

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769: 1-6, 11-26, 27, even, 44-47
Assignment
© 2008 Pearson Addison-Wesley. All rights reserved