1. Lecture 9 Dustin Lueker

2.  Can not list all possible values with probabilities ◦ Probabilities are assigned to intervals of numbers  Probability of an individual number is 0  Probabilities have to be between 0 and 1 ◦ Probability of the interval containing all possible values equals 1 ◦ Mathematically, a continuous probability distribution corresponds to a (density) function whose integral equals 1 2STA 291 Fall 2009 Lecture 9

3. 3  Let X=Weekly use of gasoline by adults in North America (in gallons)  P(16<X<21)=0.34  The probability that a randomly chosen adult in North America uses between 16 and 21 gallons of gas per week is 0.34 STA 291 Fall 2009 Lecture 9

4. 4  Discrete Variables: ◦ Histogram ◦ Height of the bar represents the probability  Continuous Variables: ◦ Smooth, continuous curve ◦ Area under the curve for an interval represents the probability of that interval STA 291 Fall 2009 Lecture 9

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6. 6  Gaussian Distribution ◦ Carl Friedrich Gauss (1777-1855)  Perfectly symmetric and bell-shaped ◦ Empirical rule applies  Probability concentrated within 1 standard deviation of the mean is always 0.68  Probability concentrated within 2 standard deviations of the mean is always 0.95  Probability concentrated within 3 standard deviations of the mean is always 0.997  Characterized by two parameters ◦ Mean = μ ◦ Standard Deviation = σ STA 291 Fall 2009 Lecture 9

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8. 8  Assume that adult female height has a normal distribution with mean μ=165 cm and standard deviation σ=9 cm ◦ With probability 0.68, a randomly selected adult female has height between μ - σ = 156 cm and μ + σ = 174 cm  This means that on the normal distribution graph of adult female heights the area under the curve between 156 and 174 is.68 ◦ With probability 0.95, a randomly selected adult female has height between μ - 2σ = 147 cm and μ + 2σ = 183 cm  This means that on the normal distribution graph of adult female heights the area under the curve between 147and 183 is.95 STA 291 Fall 2009 Lecture 9

9. 9  So far, we have looked at the probabilities within one, two, or three standard deviations from the mean using the Empirical Rule (μ + σ, μ + 2σ, μ + 3σ)  How much probability is concentrated within 1.43 standard deviations of the mean?  More general, how much probability is concentrated within any number (say z) of standard deviations of the mean? STA 291 Fall 2009 Lecture 9

10. 10  Table 3 (page B-8) shows for different values of z the probability between 0 and μ + zσ ◦ Probability that a normal random variable takes any value between the mean and z standard deviations above the mean ◦ Example  z =1.43, the tabulated value is.4236  That is, the probability between 0 and 1.43 of the standard normal distribution equals.4236  Symmetry  z = -1.43, the tabulated value is.4236  That is, the probability between -1.43 and 0 of the standard normal distribution equals.4236  So, within 1.43 standard deviations of the mean is how much probability? STA 291 Fall 2009 Lecture 9

11. 11  P(-1<Z<1) should be about 68% ◦ P(-1<Z<1) = P(-1<Z<0) + P(0<Z<1) = 2*P(0<Z<1) = ?  P(-2<Z<2) should be about 95% ◦ P(-2<Z<2) = P(-2<Z<0) + P(0<Z<2) = 2*P(0<Z<2) = ?  P(-3<Z<3) should be about 99.7% ◦ P(-3<Z<3) = P(-3<Z<0) + P(0<Z<3) = 2*P(0<Z<3) = ? STA 291 Fall 2009 Lecture 9

12. 12  We can also use the table to find z-values for given probabilities  Find the z-value corresponding to a right- hand tail probability of 0.025  This corresponds to a probability of 0.475 between 0 and z standard deviations  Table: z = 1.96 ◦ P(Z>1.96) =.025 STA 291 Fall 2009 Lecture 9

13. 13  For a normal distribution, how many standard deviations from the mean is the 90 th percentile? ◦ What is the value of z such that 0.90 probability is less than z?  P(Z<z) =.90 ◦ If 0.9 probability is less than z, then there is 0.4 probability between 0 and z  Because there is 0.5 probability less than 0  This is because the entire curve has an area under it of 1, thus the area under half the curve is 0.5  z=1.28  The 90 th percentile of a normal distribution is 1.28 standard deviations above the mean STA 291 Fall 2009 Lecture 9

14. 14  What is the z-value such that the probability is 0.1 that a normally distributed random variable falls more than z standard deviations above or below the mean?  Symmetry ◦ We need to find the z-value such that the right-tail probability is 0.05 (more than z standard deviations above the mean) ◦ z=1.65 ◦ 10% probability for a normally distributed random variable is outside 1.65 standard deviations from the mean, and 90% is within 1.65 standard deviations from the mean  Find the z-value such that the probability is 0.5 that a normally distributed random variable falls more than z standard deviations above or below the mean STA 291 Fall 2009 Lecture 9