Understanding Normal Distribution and Calculating Z-Scores

1. 5-Minute Check on Activity 7-10 • State the Empirical Rule: • What is the shape of a normal distribution? • Compute a z-score for x = 14, if μ = 10 and σ = 2 • What does a z-score represent? • Which will have a taller distribution: one with σ = 2or 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 Click the mouse button or press the Space Bar to display the answers.

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

3. 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

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

5. 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

6. 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 2σ2 1 y = -------- e 4√2π -(x – 16)2 2∙42

7. 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,)

8. 1.68 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 Enter Enter Read

9. a a a b Obtaining Area under Standard Normal Curve

10. 12 18 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 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%

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

12. a Example 2 Determine the area under the standard normal curve that lies to the right of • Z = -3.49 • 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

13. a b Example 3 Find the indicated probability of the standard normal random variable Z • 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

14. Using Your TI-calculator • Press 2ndVARS (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)

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

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

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

18. 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

19. 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