Chapter 6 The Normal Distribution Normal Distributions Bell Curve Area under entire curve = 1 or...
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Transcript of Chapter 6 The Normal Distribution Normal Distributions Bell Curve Area under entire curve = 1 or...
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Slide 2 Chapter 6 The Normal Distribution Slide 3 Normal Distributions Bell Curve Area under entire curve = 1 or 100% Mean = Median This means the curve is symmetric Slide 4 Normal Distributions Two parameters Mean (pronounced meeoo) Locates center of curve Splits curve in half Shifts curve along x-axis Standard deviation (pronounced sigma) Controls spread of curve Smaller makes graph tall and skinny Larger makes graph flat and wide Ruler of distribution Write as N(,) Slide 5 Slide 6 Standard Normal (Z) World Perfectly symmetric Centered at zero Half numbers below the mean and half above. Total area under the curve is 1. Can fill in as percentages across the curve. Slide 7 Standard Normal Distribution Puts all normal distributions on same scale z has center (mean) at 0 z has spread (standard deviation) of 1 Slide 8 Standard Normal Distribution z = # of standard deviations away from mean Negative z, number is below the mean Positive z, number is above the mean Written as N(0,1) Slide 9 Portal from X-world to Z-world z has no units (just a number) Puts variables on same scale Center (mean) at 0 Spread (standard deviation) of 1 Does not change shape of distribution Slide 10 Standardizing Variables z = # of standard deviations away from mean Negative z number is below mean Positive z number is above mean Slide 11 Standardizing Y ~ N(70,3). Standardize y = 68. y = 68 is 0.67 standard deviations below the mean Slide 12 Your Table is Your Friend Get out your book and find your Z-table. Look for a legend at the top of the table. Which way does it fill from? Find the Z values. Find the middle of the table. These are the areas or probabilities as you move across the table. Notice they are 50% in the middle and 100% at the end. Slide 13 Areas under curve Another way to find probabilities when values are not exactly 1, 2, or 3 away from is by using the Normal Values Table Gives amount of curve below a particular value of z z values range from 3.99 to 3.99 Row ones and tenths place for z Column hundredths place for z Slide 14 Finding Values What percent of a standard Normal curve is found in the region Z < -1.50? P(Z < 1.50) Find row 1.5 Find column.00 Value = 0.0668 Slide 15 Finding Values P(Z < 1.98) Find row 1.9 Find column.08 Value = 0.9761 Slide 16 Finding values What percent of a std. Normal curve is found in the region Z >-1.65? P(Z > -1.65) Find row 1.6 Find column.05 Value from table = 0.0495 P(Z > -1.65) = 0.9505 Slide 17 Finding values P(Z > 0.73) Find row 0.7 Find column.03 Value from table = 0.7673 P(Z > 0.73) = 0.2327 Slide 18 Finding values What percent of a std. Normal curve is found in the region 0.5 < Z < 1.4? P(0.5 < Z < 1.4) Table value 1.4 = 0.9192 Table value 0.5 = 0.6915 P(0.5 < Z < 1.4) = 0.9192 0.6915 = 0.2277 Slide 19 Finding values P(2.3 < Z < 0.05) Table value 0.05 = 0.4801 Table value 2.3 = 0.0107 P(2.3 < Z < 0.05) = 0.4801 0.0107 = 0.4694 Slide 20 Finding values Above what z-value do the top 15% of all z-value lie, i.e. what value of z cuts offs the highest 15%? P(Z > ?) = 0.15 P(Z < ?) = 0.85 z = 1.04 Slide 21 Finding values Between what two z-values do the middle 80% of the obs lie, i.e. what values cut off the middle 80%? Find P(Z < ?) = 0.10 Find P(Z < ?) = 0.90 Must look inside the table P(Z Solving Problems What percent of men are taller than 74 inches? P(Y > 74) = 1-P(Y Solving Problems What would you have to score to be in the top 5% of people taking the SAT verbal? P(X > ?) = 0.05? P(X < ?) = 0.95? Slide 30 Solving Problems P(Z < ?) = 0.95? z = 1.645 x is 1.645 standard deviations above mean x is 1.645(100) = 164.5 points above mean x = 500 + 164.5 = 664.5 SAT verbal score: at least 670 Slide 31 Solving Problems Between what two scores would the middle 50% of people taking the SAT verbal be? P(x 1 = ? < X < x 2 =?) = 0.50? P(-0.67 < Z < 0.67) = 0.50 x 1 = (-0.67)(100)+500 = 433 x 2 = (0.67)(100)+500 = 567 Slide 32 Solving Problems Cereal boxes are labeled 16 oz. The boxes are filled by a machine. The amount the machine fills is normally distributed with mean 16.3 oz and standard deviation 0.2 oz. Slide 33 Solving Problems What is the probability a box of cereal is underfilled? Underfilling means having less than 16 oz. P(Y < 16) = P(Z< ) = P(Z< -1.5) = 0.0668 Slide 34 Moving to Random Sample Even when we assume a variable is normally distributed if we take a random sample we need to adjust our formula slightly. Slide 35 Determining Normality If you need to determine normality I want you to use your calculators to make a box plot and to look for symmetry.