Presentation on theme: "Quantitative Methods Topic 5 Probability Distributions"— Presentation transcript:
1Quantitative Methods Topic 5 Probability Distributions
2Outline Probability Distributions Concept of making inference For categorical variablesFor continuous variablesConcept of making inference
3Reading Chapters 4, 5 and Chapter 6 (particularly Chapter 6) Fundamentals of Statistical Reasoning in Education,Colardarci et al.
4Tossing 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.
5Tossing 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……
6Do an experiment in EXCEL See animated demoCoinToss1_demo.swf
9Some 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
10Probability distribution (function) Shows the frequency (or chance) or occurrence of each value of the random variable.
11Probability 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
12Probability Distribution of Coin Toss - 2 Theoretical probabilities
13How 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).
14How 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.
15Random 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?
17Normal 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
18Standard normal distribution - 1 5% in this region2.5% in this region-1.641.96
19Standard 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).
20Standard normal distribution - 3 Around 95% of the observations lie within ± two standard deviations (strictly, ±1.96)95% in this region
21Standard normal distribution - 3 Around 68% of the observations lie within ± one standard deviation68% in this region
22Computing normal probabilities in EXCEL See animated demoNormalProbability_demo.swf
23Exercise - 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
24Exercise - 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