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Quantitative Methods

Topic 5Probability Distributions

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Outline

Probability Distributions For categorical variables For continuous variables

Concept of making inference

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Reading

Chapters 4, 5 and Chapter 6(particularly Chapter 6)

Fundamentals of Statistical Reasoning in Education,

Colardarci et al.

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Tossing a coin 10 times - 1

If the coin is not biased, we would expect “heads” to turn up 50% of the time.

However, in 10 tosses, we will not get exactly 5 “heads”.Sometimes, it could be 4 heads out of 10

tosses. Sometimes it could be 3 heads, etc.

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Tossing a coin 10 times - 2

What is the probability of getting No ‘heads’ in 10 tosses1 ‘head’ in 10 tosses2 ‘heads’ in 10 tosses3 ‘heads’ in 10 tosses……

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Do an experiment in EXCEL

See animated demo CoinToss1_demo.swf

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Frequencies of 50 sets of coin tosses

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Histogram of 50 sets of coin tosses

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Some terminology Random variable

A variable the values of which are determined by chance.

Examples of random variablesNumber of heads in 10 tosses of a coinTest score of studentsHeight Income

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Probability distribution (function)

Shows the frequency (or chance) or occurrence of each value of the random variable.

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Probability Distribution of Coin Toss - 1 Slide 8 shows the

empirical probability distribution.

Theoretical one can be computed

See animated demoBinomial Probability_demo.swf

Number of heads in 10

tosses Probability

0 0.001

1 0.010

2 0.044

3 0.117

4 0.205

5 0.246

6 0.205

7 0.117

8 0.044

9 0.010

10 0.001

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Probability Distribution of Coin Toss - 2

0.000

0.050

0.100

0.150

0.200

0.250

0.300

0 1 2 3 4 5 6 7 8 9 10

Theoretical probabilities

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How can we use the probability distribution - 1?

Provide information about “central tendency” (where the middle is, typically captured by Mean or Median), and variation (typically captured by standard deviation).

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How can we use the probability distribution - 2? Use the distribution as a point of reference Example:

If we find that, 20% of the time, we obtain only 1 head in 10 coin tosses, when the theoretical probability is about 1%, we may conclude that the coin is biased (not 50-50 chance of tossing a head)

Theoretical distribution will be better than empirical distribution, because of fluctuation in the collection of data.

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Random variables that are continuous Collect a sample of height measurement

of people. Form an empirical probability distribution Typically, the probability distribution will be

a bell-shaped curve. Compute mean and standard devation Empirical distribution is obtained Can we obtain theoretical distribution?

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Normal distribution - 1

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

-4 -3 -2 -1 0 1 2 3 4

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Normal distribution - 2

A random variable, X, that has a normal distribution with mean and standard deviation can be transformed to a variable, Z, that has standard normal distribution where the mean is 0 and the standard deviation is 1.

z-score

Need only discuss properties of the standard normal distribution

x

z

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Standard normal distribution - 1

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

-4 -3 -2 -1 0 1 2 3 4

2.5% in this

region

5% in this region

1.96-1.64

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Standard normal distribution - 2

2.5% outside 1.96 So around 5% less than -1.96, or greater than

1.96. So the general statement that

Around 95% of the observations are within -2 and 2.

More generally, around 95% of the observations are within -2 and 2 (± 2 standard deviations).

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Standard normal distribution - 3

Around 95% of the observations lie within ± two standard deviations (strictly, ±1.96)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

-4 -3 -2 -1 0 1 2 3 4

95% in this

region

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Standard normal distribution - 3

Around 68% of the observations lie within ± one standard deviation

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

-4 -3 -2 -1 0 1 2 3 4

68% in this

region

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Computing normal probabilities in EXCEL See animated demo

NormalProbability_demo.swf

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Exercise - 1

For the data set distributed in Week 2, TIMSS2003AUS,sav, for the variable bsmmat01 (second last variable, maths estimated ability),

compute the score range where the middle 95% of the scores lie: Use the observed scores and compute the percentiles

from the observations Assume the population is normally distributed

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Exercise - 2

Dave scored 538. What percentage of students obtained scores higher than Dave?Use the observed scores and compute the

percentiles from the observationsAssume the population is normally distributed