Presentation on theme: "1 Quantitative Methods Topic 5 Probability Distributions."— Presentation transcript:
1 Quantitative Methods Topic 5 Probability Distributions
2 Outline Probability Distributions 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 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 Slide 8 shows the empirical probability distribution. Theoretical one can be computed See animated demo Binomial Probability_demo.swf Number of heads in 10 tossesProbability
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 % in this region 5% in this region
19 Standard normal distribution % outside 1.96 So around 5% less than -1.96, or greater than 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