STA1501_ Descriptive Statistics And Probability_ Exam Prep & Summary.

STA1501_ Descriptive Statistics And Probability_ Exam Prep & Summary. The Normal Distribution. ‘Bell Shaped’  Symmetrical  Mean, Median and Mode are Equal Location is determined by the mean, μ Spread is determined by the standard deviation, σ The random variable has an infinite theoretical range. Any normal distribution (with any mean and standard deviation combination) can be transformed into the standardized normal distribution (Z)  Need to transform X units into Z units Translation to the Standardized Normal Distribution  Translate from X to the standardized normal (the “Z” distribution) by subtracting the mean of X and dividing by its standard deviation: σ X μ Z   The Z distribution always has mean = 0 and standard deviation = 1 Example  If X is distributed normally with mean of 100 and standard deviation of 50, the Z value for X = 200 is  This says that X = 200 is two standard deviations (2 increments of 50 units) above the mean of 100. 2.0 50 200 100 σ X μ Z      Comparing X and Z units Z 100 0 2.0 200 X Note that the distribution is the same, only the scale has changed. We can express the problem in original units (X) or in standardized units (Z) (μ = 100, σ = 50) (μ = 0, σ = 1) Finding Normal Probabilities Probability is the area under the curve! a b X f(X) P(a X ≤ b) Probability is measured by the area under the curve ≤ = P(a X b) (Note that the probability of any individual value is zero) f(X) μ X Probability as Area Under the Curve 0.5 0.5 The total area under the curve is 1.0, and the curve is symmetric, so half is above the mean, half is below P(  X  )  1.0 P(μ  X  )  0.5 P(  X  μ)  0.5 Empirical Rules μ ± 1σ encloses about 68% of X’s f(X) X μ-1σ μ μ+1σ What can we say about the distribution of values around the mean? There are some general rules: σ σ 68.26% The Empirical Rule  μ ± 2σ covers about 95% of X’s  μ ± 3σ covers about 99.7% of X’s μ x 2σ 2σ μ x 3σ 3σ 95.44% 99.73% (continued) The Standardized Normal Table  The Cumulative Standardized Normal table in the textbook (Appendix table E.2) gives the probability less than a desired value for Z (i.e., from negative infinity to Z) 0 2.00 Z 0.9772 Example: P(Z 2.00) = 0.9772 The Standardized Normal Table The value within the table gives the probability from Z =   up to the desired Z value .9772 P(Z 2.00) = 0.9772 2.0 The row shows the value of Z to the first decimal point The column gives the value of Z to the second decimal point 2.0 . . . (continued) Z 0.00 0.01 0.02 … 0.0 0.1 General Procedure for Finding Probabilities