5-Minute Check on Activity 7-10

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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 = 2 or σ = 4.

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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.

### 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: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

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,)

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

a

a

a

b

Obtaining Area under Standard Normal Curve

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%

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

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

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

• 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)

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

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

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

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