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Unformatted text preview: GEM2900: Understanding Uncertainty & Statistical Thinking DSAP, NUS, Semester 1, 2008/2009 – 1 GEM2900: Understanding Uncertainty & Statistical Thinking Berwin Turlach statba@nus.edu.sg Department of Statistics and Applied Probability National University of Singapore Reminder GEM2900: Understanding Uncertainty & Statistical Thinking DSAP, NUS, Semester 1, 2008/2009 – 196 The fourth CA is due Friday, 14 November, at 5pm. Please read the relevant announcement on IVLE. I will instruct IVLE to send reminder emails on Thursday and Friday morning. Please submit your answers if you do not want to receive these emails. Calculating with the normal distribution (cont.) GEM2900: Understanding Uncertainty & Statistical Thinking DSAP, NUS, Semester 1, 2008/2009 – 197 squaresolid Consider a normally distributed random variable X with mean μ and variance σ 2 , and the random variable Z = X μ σ which is standard normal. squaresolid The for c > P( μ cσ < X < μ + cσ ) = P( c < Z < c ) squaresolid For example P( μ σ < X < μ + σ ) = P( 1 < Z < 1) ≈ . 682 P( μ 2 σ < X < μ + 2 σ ) = P( 2 < Z < 2) ≈ . 954 P( μ 3 σ < X < μ + 3 σ ) = P( 3 < Z < 3) ≈ . 997 squaresolid This is sometimes called the 68 — 95 — 99.7 rule. squaresolid If data are normally distributed, roughly 68% ( ≈ 2 3 ) of observations should be within one standard deviation (SD) of the mean; and roughly 95% of observations should be within two SDs of the mean. The 68 — 95 — 99.7 rule GEM2900: Understanding Uncertainty & Statistical Thinking DSAP, NUS, Semester 1, 2008/2009 – 198 Normal probability density function μ  3 σ μ  2 σ μ  σ μ μ + σ μ + 2 σ μ + 3 σ Mean μ Standard deviation σ Probability of falling into given region 68% 95% 99.7% Why is the normal distribution so important? GEM2900: Understanding Uncertainty & Statistical Thinking DSAP, NUS, Semester 1, 2008/2009 – 199 Motivating example: sums of dice squaresolid The next five slides show the probabilities of the sums of n = 1 , 2 , 3 , 4 and 5 dice. squaresolid What happens to the shape of the probabilities as the number of dice n increases? squaresolid Amazingly, this will happen with (almost) any random variable ; as long as n is large enough the probabilities for the sum (and mean) will start to follow a normal bellshaped curve. Why is the normal distribution so important? (cont.) GEM2900: Understanding Uncertainty & Statistical Thinking DSAP, NUS, Semester 1, 2008/2009 – 200 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 values Sum of one die 0.00 0.05 0.10 0.15 Why is the normal distribution so important?...
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