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**Quantitative Methods Topic 5 Probability Distributions**

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**Outline Probability Distributions Concept of making inference**

For categorical variables For continuous variables Concept of making inference

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**Reading Chapters 4, 5 and Chapter 6 (particularly Chapter 6)**

Fundamentals of Statistical Reasoning in Education, Colardarci et al.

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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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**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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**Do an experiment in EXCEL**

See animated demo CoinToss1_demo.swf

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**Frequencies of 50 sets of coin tosses**

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**Histogram of 50 sets of coin tosses**

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**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

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**Probability distribution (function)**

Shows the frequency (or chance) or occurrence of each value of the random variable.

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**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

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**Probability Distribution of Coin Toss - 2**

Theoretical probabilities

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**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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**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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**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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Normal distribution - 1

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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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**Standard normal distribution - 1**

5% in this region 2.5% in this region -1.64 1.96

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**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).

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**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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**Standard normal distribution - 3**

Around 68% of the observations lie within ± one standard deviation 68% in this region

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**Computing normal probabilities in EXCEL**

See animated demo NormalProbability_demo.swf

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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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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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