5-Minute Check on Activity 7-105-Minute Check on Activity 7-105-Minute Check on Activity 7-105-Minute Check on Activity 7-10

Click the mouse button or press the Space Bar to display the answers.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

Activity 7 - 11

Part-time JobsMcDonald’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:

where μ is the mean and σ is the standard deviation of the random variable x

In our example this gives us:

1y = -------- e √2π

-(x – μ)2

2σ2

1y = -------- e 4√2π

-(x – 16)2

2∙42

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 - μz = -------- = 1.68 Enter

Enter

Read

Enter1.68

Approach Graphically Solution

Find the area to the left of za

P(Z < a)

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.

Normcdf(-E99,a,0,1)

Find the area to the right of za

P(Z > a) or

1 – P(Z < a)

Shade 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)

Find the area between za and zb

P(a < Z < b)

Shade 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)

Obtaining Area under Standard Normal Curve

a

a

a b

Activity contMany 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%

12 18

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

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

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 -2.55is .0054

table look up for area to the left of 2.55is .9946

are between them = 0.9946 – 0.0054 = 0.98923

a 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) Z = 0.92

b) Z = 2.90

a

Normalcdf(-E99,0.92) = 0.821214

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

b) Z = 3.45

Normalcdf(2.23,E99) = 0.012874

Normalcdf(3.45,E99) = 0.00028

a

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

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