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1 Quantitative Methods Topic 5 Probability Distributions

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

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

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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.

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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 ……

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

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7 Frequencies of 50 sets of coin tosses

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8 Histogram of 50 sets of coin tosses

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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

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

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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

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12 Probability Distribution of Coin Toss - 2 Theoretical probabilities

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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).

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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.

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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?

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16 Normal distribution - 1

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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

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18 Standard normal distribution % in this region 5% in this region

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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).

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

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

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

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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

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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

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