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## Chapter 5 Normal Curve

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Title: Chapter 5 Normal Curve

1
Chapter 5 Normal Curve
• Bell Shaped
• Unimodal
• Symmetrical
• Unskewed
• Mode, Median, and Mean are same value

2
Theoretical Normal Curve
• General relationships

3
Theoretical Normal Curve
4
Using the Normal Curve Z Scores
• To find areas, first compute Z scores.
• The formula changes a raw score (Xi) to a
standardized score (Z).

5
Using Appendix A to Find Areas Below a Score
• Appendix A can be used to find the areas above
and below a score.
• First compute the Z score, taking careful note of
the sign of the score.
• Draw a picture of the normal curve and shade in
the area in which you are interested.

6
Using Appendix A
• Appendix A has three columns.
• (a) Z scores.
• (b) areas between the score and the mean

7
Using Appendix A
• Appendix A has three columns.
• ( c) areas beyond the Z score

8
Using Appendix A
• Find your Z score in Column A.
• To find area below a positive score
• Add column b area to .50.
• To find area above a positive score
• Look in column c.

(a) (b) (c)
. . .
1.66 0.4515 0.0485
1.67 0.4525 0.0475
1.68 0.4535 0.0465
. . .
9
Using Appendix A
• The area below Z 1.67 is 0.4525 0.5000 or
0.9525.
• Areas can be expressed as percentages
• 0.9525 95.25

10
Using Appendix A
• What if the Z score is negative (1.67)?
• To find area below a negative score
• Look in column c.
• To find area above a negative score

(a) (b) (c)
. . .
1.66 0.4515 0.0485
1.67 0.4525 0.0475
1.68 0.4535 0.0465
. . .
11
Using Appendix A
• The area below Z - 1.67 is 0.475.
• Areas can be expressed as 4.75.
• Areas under the curve can also be expressed as
probabilities.
• Probabilities are proportions and range from 0.00
to 1.00.
• The higher the value, the greater the probability
(the more likely the event).

12
Finding Probabilities
• If a distribution has
• 13
• s 4
• What is the probability of randomly selecting a
score of 19 or more?

13
Finding Probabilities
1. Find the Z score.
2. For Xi 19, Z 1.50.
3. Find area above in column c.
4. Probability is 0.0668 or 0.07.

(a) (b) (c)
. . .
1.49 0.4319 0.0681
1.50 0.4332 0.0668
1.51 0.4345 0.0655
. . .
14
Finding Probabilities (exercise 1)
• The mean of the grades of final papers for this
class is 65 and the standard deviation is 5. What
percentage of the students have scores above 70?
In other words, what is the probability of
randomly selecting a score of 70 or more?

15
Finding Probabilities (exercise 2)
• Stephen Jay Gould (1996). Full House. The Spread
of Excellence from Plato to Darwin.
• Doctors you have an aggressive type of cancer
and half of the patients will die within 8
months.
• Question An optimistic person like Gould was not
impressed and not shocked by this message. Why
not?

16
Chapter 6 Introduction to Inferential
Statistics Sampling and the Sampling
Distribution
• Problem The populations we wish to study are
almost always so large that we are unable to
gather information from every case.
• ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
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? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

17
Basic Logic And Terminology
• Solution We choose a sample -- a carefully
chosen subset of the population and use
information gathered from the cases in the sample
to generalize to the population.
• ? ? ?
• ? ? ? ?
• ? ? ? ? ? ? ? ?
• ?
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• ? ? ?

18
Samples
• Must be representative of the population.
• Representative The sample has the same
characteristics as the population.
• How can we ensure samples are representative?
• Samples in which every case in the population has
the same chance of being selected for the sample
are likely to be representative.

19
Sampling Techniques
• Simple Random Sampling (SRS)
• Systematic Random Sampling
• Stratified Random Sampling
• Cluster Sampling
differences between those techniques

20
Applying Logic and Terminology
• For example
• Population All 20,000 students.
• Sample The 500 students selected and
interviewed

21
The Sampling Distribution
• Every application of inferential statistics
involves 3 different distributions.
• Information from the sample is linked to the
population via the sampling distribution.

Population
Sampling Distribution
Sample
22
First Theorem
• Tells us the shape of the sampling distribution
and defines its mean and standard deviation.
• If we begin with a trait that is normally
distributed across a population (IQ, height) and
take an infinite number of equally sized random
samples from that population, the sampling
distribution of sample means will be normal.

23
Central Limit Theorem
• For any trait or variable, even those that are
not normally distributed in the population, as
sample size grows larger, the sampling
distribution of sample means will become normal
in shape.

24
The Sampling Distribution Properties
• Normal in shape.
• Has a mean equal to the population mean.
• Has a standard deviation (standard error) equal
to the population standard deviation divided by
the square root of N.
• The Sampling Distribution is normal so we can use
Appendix A to find areas.
• See Table 6.1, p. 160 of Healeys book for
specific important symbols.