Chapter 5 Normal Curve

- Bell Shaped
- Unimodal
- Symmetrical
- Unskewed
- Mode, Median, and Mean are same value

Theoretical Normal Curve

- General relationships
- 1 s about 68
- 2 s about 95
- 3 s about 99

Theoretical Normal Curve

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).

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.

Using Appendix A

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

Using Appendix A

- Appendix A has three columns.
- ( c) areas beyond the Z score

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

. . .

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

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

. . .

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).

Finding Probabilities

- If a distribution has
- 13
- s 4
- What is the probability of randomly selecting a

score of 19 or more?

Finding Probabilities

- Find the Z score.
- For Xi 19, Z 1.50.
- Find area above in column c.
- 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

. . .

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?

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?

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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? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

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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- ? ? ?

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.

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

Applying Logic and Terminology

- For example
- Population All 20,000 students.
- Sample The 500 students selected and

interviewed

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

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.

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.

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.