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
• 1 s about 68
• 2 s about 95
• 3 s about 99

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
• Add column b .50

(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
• See Healeys book for more information on
  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.