#### Transcript Activity 11

5-Minute Check on Activity 7-10 1. State the Empirical Rule: Also known as 68-95-99.7 rule (± nσ’s from μ) 2. What is the shape of a normal distribution? Symmetric mound-like 3. Compute a z-score for x = 14, if μ = 10 and σ = 2 Z = (14-10)/2 = 2 4. What does a z-score represent? Number of standard deviations away from the mean 5. Which will have a taller distribution: one with σ = 2 or σ = 4 Larger spread is smaller height; so σ = 2 is taller Click the mouse button or press the Space Bar to display the answers. Activity 7 - 11 Part-time Jobs McDonald’s Times Square, New York, NY, 1/3/2009 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 Vocabulary • Cumulative Probability Density Function – the sum of the area under a density curve from the left 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 Normal Probability Density Function There is a y = f(x) style function that describes the normal curve: 1 -(x – μ)2 y = -------- e 2σ2 √2π where μ is the mean and σ is the standard deviation of the random variable x In our example this gives us: 1 -(x – 16)2 y = -------- e 2∙42 4√2π 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,) 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 1.68 Enter Read Obtaining Area under Standard Normal Curve Approach Graphically Solution Shade the area to the left of za Use Table IV to find the row and column that correspond to za. The area is the value where the row and column intersect. Find the area to the left of za P(Z < a) Normcdf(-E99,a,0,1) a Shade the area to the right of za Find the area to the right of za Use Table IV to find the area to the left of za. The area to the right of za is 1 – area to the left of za. Normcdf(a,E99,0,1) or 1 – Normcdf(-E99,a,0,1) P(Z > a) or 1 – P(Z < a) a Shade the area between za and zb Find the area between za and zb Use Table IV to find the area to the left of za and to the left of za. The area between is areazb – areaza. Normcdf(a,b,0,1) P(a < Z < b) a b 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 12 18 z12 = (12-16)/4 = -1 and z18 = (18-16)/4 = 0.5 Using Appendix C: P(z0.5)= 0.6915 and p(z-1)=0.1587 So P(12 < x < 18) = 0.6915 – 0.1587 = .5328 or 53.28% Example 1 Determine the area under the standard normal curve that lies to the left of a a) Z = -3.49 table look up yields: 0.0002 b) Z = 1.99 table look up yields: 0.9767 Example 2 Determine the area under the standard normal curve that lies to the right of a) Z = -3.49 a table look up yields: .0002 to the left of -3.49 area to the right = 1 – 0.0002 = 0.9998 b) Z = -0.55 table look up yields: .2912to the left of -0.55 area to the right = 1 – 0.2912 = 0.70884 Example 3 Find the indicated probability of the standard normal random variable Z a 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 b Using Your TI-calculator • Press 2nd 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) Example 4 Determine the area under the standard normal curve that lies to the left of a a) Z = 0.92 Normalcdf(-E99,0.92) = 0.821214 b) Z = 2.90 Normalcdf(-E99,2.90) = 0.998134 Example 5 Determine the area under the standard normal curve that lies to the right of a) Z = 2.23 Normalcdf(2.23,E99) = 0.012874 b) Z = 3.45 Normalcdf(3.45,E99) = 0.00028 a Example 6 Find the indicated probability of the standard normal random variable Z a a) P(-0.55 < Z < 0) Normalcdf(-0.55,0) = 0.20884 b) P(-1.04 < Z < 2.76) Normalcdf(-1.04,2.76) = 0.84794 b 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 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