Presentation on theme: "Quantitative Methods Topic 5 Probability Distributions"— Presentation transcript:
1 Quantitative Methods Topic 5 Probability Distributions
2 Outline Probability Distributions Concept of making inference For categorical variablesFor continuous variablesConcept 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 - 1If 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 tosses1 ‘head’ in 10 tosses2 ‘heads’ in 10 tosses3 ‘heads’ in 10 tosses……
6 Do an experiment in EXCEL See animated demoCoinToss1_demo.swf
9 Some terminology Random variable Examples of random variables A variable the values of which are determined by chance.Examples of random variablesNumber of heads in 10 tosses of a coinTest score of studentsHeightIncome
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 tossesProbability0.00110.01020.04430.11740.20550.246678910Slide 8 shows the empirical probability distribution.Theoretical one can be computedSee animated demoBinomial 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 referenceExample: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 distributionTypically, the probability distribution will be a bell-shaped curve.Compute mean and standard devationEmpirical distribution is obtainedCan we obtain theoretical distribution?
17 Normal distribution - 2A 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-scoreNeed only discuss properties of the standard normal distribution
18 Standard normal distribution - 1 5% in this region2.5% in this region-1.641.96
19 Standard normal distribution - 2 2.5% outside 1.96So around 5% less than -1.96, or greater than 1.96.So the general statement thatAround 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 deviation68% in this region
22 Computing normal probabilities in EXCEL See animated demoNormalProbability_demo.swf
23 Exercise - 1For 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 observationsAssume the population is normally distributed
24 Exercise - 2Dave 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