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5-Minute Check on Activity 7-10 Click the mouse button or press the Space Bar to display the answers. 1.State the Empirical Rule: 2.What is the shape of a normal distribution? 3.Compute a z-score for x = 14, if μ = 10 and σ = 2 4.What does a z-score represent? 5.Which will have a taller distribution: one with σ = 2 or σ = 4 Also known as 68-95-99.7 rule (± nσ’s from μ) Symmetric mound-like Z = (14-10)/2 = 2 Number of standard deviations away from the mean Larger spread is smaller height; so σ = 2 is taller

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Activity 7 - 11 Part-time Jobs McDonald’s Times Square, New York, NY, 1/3/2009

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Objectives Determine the area under the standard normal curve using the z-table Standardize a normal curve Determine the area under the standard normal curve using a calculator

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Vocabulary Cumulative Probability Density Function – the sum of the area under a density curve from the left

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Activity Many high school students have part-time jobs after school and on weekends. Suppose the number of hours students spend working per week is approximately normally distributed, with a mean of 16 hours and a standard deviation of 4 hours. If a student is randomly selected, what is the probability that the student works between 12 and 18 hours per week? Mean = 16 Standard Deviation (StDev) = 4 so one StDev below = 12 and ½ StDev above = 18 can use z-tables: P(12 < x < 18) = P( -1 < z < 0.5) but using calculator is much easier!: P(12 < x < 18) = normcdf(12, 18, 16, 4) = 0.5328

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Normal Probability Density Function There is a y = f(x) style function that describes the normal curve: where μ is the mean and σ is the standard deviation of the random variable x In our example this gives us: 1 y = -------- e √2π -(x – μ) 2 2σ 2 1 y = -------- e 4√2π -(x – 16) 2 2∙4 2

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Probability and Normal Curve All possible probabilities sum to 1 Normal curve is a probability density function Area under the curve will sum to 1 The area between two values is the probability that a value will occur between those two values Standard Normal is a normal curve with a mean of 0 and a standard deviation of 1 Normal notation: X ~ N(μ, )

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Z-tables Z-table: A table that gives the cumulative area under a standardized normal curve from the left to the z-value x - μ z = -------- = 1.68 Enter Read Enter 1.68

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ApproachGraphicallySolution Find the area to the left of z a P(Z < a) Shade the area to the left of z a Use Table IV to find the row and column that correspond to z a. The area is the value where the row and column intersect. Normcdf(-E99,a,0,1) Find the area to the right of z a P(Z > a) or 1 – P(Z < a) Shade the area to the right of z a Use Table IV to find the area to the left of z a. The area to the right of z a is 1 – area to the left of z a. Normcdf(a,E99,0,1) or 1 – Normcdf(-E99,a,0,1) Find the area between z a and z b P(a < Z < b) Shade the area between z a and z b Use Table IV to find the area to the left of z a and to the left of z a. The area between is area zb – area za. Normcdf(a,b,0,1) Obtaining Area under Standard Normal Curve aa a b

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Activity cont Many high school students have part-time jobs after school and on weekends. Suppose the number of hours students spend working per week is approximately normally distributed, with a mean of 16 hours and a standard deviation of 4 hours. If a student is randomly selected, what is the probability that the student works between 12 and 18 hours per week? We want so we convert 12 and 18 to z-values z 12 = (12-16)/4 = -1 and z 18 = (18-16)/4 = 0.5 Using Appendix C: P(z 0.5 )= 0.6915 and p(z -1 )=0.1587 So P(12 < x < 18) = 0.6915 – 0.1587 =.5328 or 53.28% 12 18

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Example 1 Determine the area under the standard normal curve that lies to the left of a)Z = -3.49 b)Z = 1.99 a table look up yields: 0.0002 table look up yields: 0.9767

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Example 2 Determine the area under the standard normal curve that lies to the right of a)Z = -3.49 b)Z = -0.55 table look up yields:.0002 to the left of -3.49 area to the right = 1 – 0.0002 = 0.9998 table look up yields:.2912to the left of -0.55 area to the right = 1 – 0.2912 = 0.70884 a

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Example 3 Find the indicated probability of the standard normal random variable Z a)P(-2.55 < Z < 2.55) table look up for area to the left of -2.55 is.0054 table look up for area to the left of 2.55 is.9946 are between them = 0.9946 – 0.0054 = 0.98923 a b

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Using Your TI-calculator Press 2 nd VARS (DISTR menu) Press 2 (normalcdf) Parameters Required: –Left value –Right value –Mean, μ –Standard Deviation, Using your calculator, normcdf(left, right, μ, σ) Notes: –Use –E99 for negative infinity –Use E99 for positive infinity –Don’t have to plug in 0,1 for μ, (it assumes standard normal)

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Example 4 Determine the area under the standard normal curve that lies to the left of a)Z = 0.92 b)Z = 2.90 a Normalcdf(-E99,0.92) = 0.821214 Normalcdf(-E99,2.90) = 0.998134

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Example 5 Determine the area under the standard normal curve that lies to the right of a)Z = 2.23 b)Z = 3.45 Normalcdf(2.23,E99) = 0.012874 Normalcdf(3.45,E99) = 0.00028 a

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Example 6 Find the indicated probability of the standard normal random variable Z a)P(-0.55 < Z < 0) b)P(-1.04 < Z < 2.76) Normalcdf(-0.55,0) = 0.20884 Normalcdf(-1.04,2.76) = 0.84794 a b

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Finding Area under any Normal Curve Draw a normal curve and shade the desired area Use your calculator, normcdf(left, right, μ, σ) OR Convert the x-values to Z-scores using Z = (x – μ) / σ Draw a standard normal curve and shade the area desired Find the area under the standard normal curve using the table. This area is equal to the area under the normal curve drawn in Step 1

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Summary and Homework Summary –Normal Curve Properties Area under a normal curve sums to 1 Area between two points under the normal curve represents the probability of x being between those two points –Standard Normal Curves Appendix C has z-tables for cumulative areas Calculator can find the area quicker and easier –TI-83 Help for Normalcdf(LB,UB, , ) LB is lower bound; UB is upper bound is the mean and is the standard deviation Homework –pg 881-883; problems 1, 3-5

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