1. Quantitative Methods Topic 5 Probability Distributions

2. Outline Probability Distributions Concept of making inference
For categorical variables
For continuous variables
Concept of making inference

3. Reading Chapters 4, 5 and Chapter 6 (particularly Chapter 6)
Fundamentals of Statistical Reasoning in Education,
Colardarci et al.

4. 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.

5. Tossing a coin 10 times - 2 What is the probability of getting
No ‘heads’ in 10 tosses
1 ‘head’ in 10 tosses
2 ‘heads’ in 10 tosses
3 ‘heads’ in 10 tosses ……

6. Do an experiment in EXCEL
See animated demo
CoinToss1_demo.swf

7. Frequencies of 50 sets of coin tosses

8. Histogram of 50 sets of coin tosses

9. Some terminology Random variable Examples of random variables
A variable the values of which are determined by chance.
Examples of random variables
Number of heads in 10 tosses of a coin
Test score of students
Height
Income

10. Probability distribution (function)
Shows the frequency (or chance) or occurrence of each value of the random variable.

11. Probability Distribution of Coin Toss - 1
Number of heads in 10 tosses
Probability
0.001 1
0.010 2
0.044 3
0.117 4
0.205 5
0.246 6 7 8 9 10
Slide 8 shows the empirical probability distribution.
Theoretical one can be computed
See animated demo
Binomial Probability_demo.swf

12. Probability Distribution of Coin Toss - 2
Theoretical probabilities

13. 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).

14. 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 chance of tossing a head)
Theoretical distribution will be better than empirical distribution, because of fluctuation in the collection of data.

15. 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?

16. Normal distribution - 1

17. 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

18. Standard normal distribution - 1
5% in this region
2.5% in this region
-1.64
1.96

19. 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).

20. Standard normal distribution - 3
Around 95% of the observations lie within ± two standard deviations (strictly, ±1.96)
95% in this region

21. Standard normal distribution - 3
Around 68% of the observations lie within ± one standard deviation
68% in this region

22. Computing normal probabilities in EXCEL
See animated demo
NormalProbability_demo.swf

23. 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

24. Exercise - 2
Dave scored 538. What percentage of students obtained scores higher than Dave?
Use the observed scores and compute the percentiles from the observations
Assume the population is normally distributed