Standard Normal Standard Normal CalculationsCalculations

2.2 b cont.2.2 b cont. Target Goal: I can standardize individual Target Goal: I can standardize individual

values and compare them using a common values and compare them using a common scalescale

Hw: pg 105: 13 – 15,Hw: pg 105: 13 – 15,

pg 132: 54ab(see 53 as example), 55, 56, 58, 59pg 132: 54ab(see 53 as example), 55, 56, 58, 59 ..

Standardizing and Z-ScoresStandardizing and Z-Scores

All normal curves are the same if we measure All normal curves are the same if we measure in units of size σ about the mean μ as center. in units of size σ about the mean μ as center. Changing to these units is called Changing to these units is called standardizingstandardizing..

If x is an observation from a distribution that has mean If x is an observation from a distribution that has mean μ and standard deviation σ, then the standardized μ and standard deviation σ, then the standardized value, called the value, called the z-scorez-score of x is:of x is:

xz

Standardizing and Z-ScoresStandardizing and Z-Scores

A A z-scorez-score tells us tells us how many standard how many standard deviationsdeviations the original observationthe original observation falls falls away from the meanaway from the mean and inand in which which direction.direction.

Observations larger than the mean are Observations larger than the mean are positivepositive..

Observations smaller than the mean are Observations smaller than the mean are negativenegative..

Standard Normal DistributionStandard Normal Distribution

If a variable x has any normal dist. N(μ,σ),If a variable x has any normal dist. N(μ,σ), Then the new standardized variable Then the new standardized variable

producedproduced

has the standard normal distribution.has the standard normal distribution.

OriginalOriginal StandardizeStandardize NewNew

xx, N(μ,σ) , N(μ,σ) z z z, z, N(0,1)N(0,1)

xz

Ex 1: Standardizing Women’s HeightsEx 1: Standardizing Women’s Heights

The heights of young women are approx. The heights of young women are approx. normal with normal with μ = 64.5 μ = 64.5 inches and inches and σ = 2.5 σ = 2.5 inches.inches.

The standardized height is:The standardized height is: z = (z = (height – )/height – )/

A women’s A women’s standardized heightstandardized height is the is the number of number of standardstandard deviations deviations by which by which her height her height differsdiffers from the from the mean height mean height of all young womenof all young women..

64.5 2.5

μ = μ = 64.564.5 inches and σ = inches and σ = 2.52.5

A women A women 6868 inches has a standardized inches has a standardized height ?height ?

z = ( – )/ = z = ( – )/ =

standard deviations standard deviations above the mean.above the mean. A women A women 5 feet 5 feet tall has a standardized tall has a standardized

height?height?

standard deviations standard deviations below the mean.below the mean.

68

64.5 2.5

z = (60 – 64.5)/2.5 = -1.8

1.4

StandardizingStandardizing

Standardizing gives a Standardizing gives a common scalecommon scale and produces a and produces a new variablenew variable that has that has the standard normal distribution.the standard normal distribution.

Exercise 2: Comparing Batting AveragesExercise 2: Comparing Batting Averages

Three landmarks of baseball achievement are Three landmarks of baseball achievement are Ty Cobb’s batting average of Ty Cobb’s batting average of .420.420 in in 19111911, Ted , Ted Williams Williams .406.406 in in 19411941, and George Brett’s , and George Brett’s .390.390 in in 19801980. .

These These batting averages cannot be compared batting averages cannot be compared directlydirectly because the distribution of major league because the distribution of major league batting averages has changed over the years.batting averages has changed over the years.

While the mean batting average has been held While the mean batting average has been held roughly constant by roughly constant by rule changes and balance rule changes and balance between hitting and pitchingbetween hitting and pitching, the standard , the standard deviation has dropped over time. deviation has dropped over time.

Here are the facts:Here are the facts:

DecadeDecade MeanMean Std. Dev Std. Dev1910’s1910’s .266.266 .0371.03711940’s1940’s .267.267 .0326.03261970’s1970’s .261.261 .0317.0317

Compute the standardized batting Compute the standardized batting averages for Cobb, Williams, and Brett to averages for Cobb, Williams, and Brett to compare how far each stood above his compare how far each stood above his peers.peers.

Who was the best among his peers?Who was the best among his peers?

Cobb(.420 ); Cobb(.420 ); 1910’s1910’s (.266,.0371) : (.266,.0371) :

z =z =

z = 4.15z = 4.15

Wiliams(.406 ); Wiliams(.406 ); 1940’s1940’s(.267,.0326):(.267,.0326):

z = 4.26z = 4.26 Brett(.390 ); Brett(.390 ); 1970’s1970’s(.261,.0317): (.261,.0317):

z = 4.07z = 4.07

Williams z-score is the highest.Williams z-score is the highest.

xz

.266 .420 -

.0371

Normal Distribution CalculationsNormal Distribution Calculations

The area under a density curve is The area under a density curve is a a proportion of the observations in a proportion of the observations in a distribution.distribution.

After standardizing, all normal After standardizing, all normal distributions are distributions are the same.the same.

Table ATable A gives areas under the curve gives areas under the curve for the standard normal distribution.for the standard normal distribution.

The table entry for each The table entry for each z value z value is the area is the area under the curve to the left of under the curve to the left of z.z.

Be careful if the problem asks for Be careful if the problem asks for the area to the the area to the left or to the right of the z value.left or to the right of the z value.

Always sketch the normal curve, mark the z Always sketch the normal curve, mark the z value, and shade the area of interest.value, and shade the area of interest.

Ex. 3 Using the Ex. 3 Using the ZZ Table Table

Find the proportion of observations from the Find the proportion of observations from the standard normal distribution that are less than standard normal distribution that are less than 1.4.1.4.

Look in Table A to verify Look in Table A to verify

Ex 1. Cont. Women’s HeightEx 1. Cont. Women’s Height

What proportion of all young women are less What proportion of all young women are less than 68 inches tall. (We already calculated z).than 68 inches tall. (We already calculated z).

The The area to the left of z = 1.4 under the standard area to the left of z = 1.4 under the standard normal curvenormal curve is the same as the is the same as the area to the left area to the left of x = 68. (Use Table A)of x = 68. (Use Table A)

1.4x

z

The proportion is 91.92%.

Finding Normal ProportionsFinding Normal Proportions

1.1. State: State: the problemthe problem in terms of the in terms of the observed observed variable variable xx.. Draw a picture of the distribution Draw a picture of the distribution and shade the area of interest under the curve.and shade the area of interest under the curve.

2.2. Plan:Plan: Standardize Standardize xx and restate the problem and restate the problem in terms of a standard normal variable in terms of a standard normal variable zz. Draw . Draw a picture to show the area of interest under the a picture to show the area of interest under the standard normal curve.standard normal curve.

3.3. DO: DO: Find the required area Find the required area under the under the standard normal curve, standard normal curve, using Table Ausing Table A and the and the fact that the total area under the curve is 1.fact that the total area under the curve is 1.

4.4. Conclude:Conclude: Write your conclusion Write your conclusion in the in the context of the problem.context of the problem.

Exercise 4: Table A PracticeExercise 4: Table A Practice

Use Table A. In each case, sketch a Use Table A. In each case, sketch a standard normal curve and shade the standard normal curve and shade the area under the curve that is the answer area under the curve that is the answer to the question.to the question.

a.a. z < 2.85z < 2.85

b.b. z > 2.85z > 2.85

c.c. z > -1.66z > -1.66

d.d. -1.66 < z < 2.85-1.66 < z < 2.85

.9978

1-.9978 = .0022

1 - .0485 = .9515

.9978 -.0485 = .9493

Exercise 5: Heights of American MenExercise 5: Heights of American Men

The distribution of adult American men is The distribution of adult American men is approximately normal with mean 69 approximately normal with mean 69 inches and standard deviation 2.5 inches and standard deviation 2.5 inches.inches.

a.a. What percent of men are What percent of men are at least 6 feetat least 6 feet (72 inches) tall?(72 inches) tall?

Step 1: Step 1: State the problem.State the problem.

We want the proportion of men’s heights We want the proportion of men’s heights with x with x ≥≥ 72 inches. 72 inches.

Step 2: PlanStep 2: Plan

mean 69, standard deviation 2.5 inchesmean 69, standard deviation 2.5 inches

Step 2: SStep 2: Standardize and draw a picture.tandardize and draw a picture.

x x ≥≥ 72 72

(x - μ)/σ (x - μ)/σ ≥≥ ( - )/ ( - )/

z z ≥≥ 1.2 1.2

72

69 2.5

z z ≥≥ 1.2 1.2

Step 3: Do -Step 3: Do -Use the table.Use the table.

Area is: 1 – P( z < 1.2 ) = Area is: 1 – P( z < 1.2 ) =

1 – 0.8849 = 0.11511 – 0.8849 = 0.1151

Step 4: Conclude in cStep 4: Conclude in context.ontext.

About About 11.5% 11.5% of all adult men are at least 6 of all adult men are at least 6 feet tall.feet tall.

b. What percent of men are between 5 feet (60 b. What percent of men are between 5 feet (60 inches) and 6 feet?inches) and 6 feet?

Step 1: Step 1: State the problem.State the problem.

We want the proportion of men’s heights We want the proportion of men’s heights with 60 ≤ x ≤ 72 inches. with 60 ≤ x ≤ 72 inches.

Step 2: Plan - SStep 2: Plan - Standardize and draw a tandardize and draw a picture.picture.

mean 69, standard deviation 2.5 inchesmean 69, standard deviation 2.5 inches 60 ≤ x ≤ 72.60 ≤ x ≤ 72. 60 ≤ 60 ≤ (x - μ)/σ(x - μ)/σ ≤ 72 ≤ 72 ( - )/ ≤ ( - )/ ≤ zz ≤ ( - )/ ≤ ( - )/

(60 - 69 )/ 2.5 (60 - 69 )/ 2.5 ≤≤ zz ≤≤ ( 72 - 69 )/2.5( 72 - 69 )/2.5 -3.6 ≤ z ≤-3.6 ≤ z ≤ 1.2 1.2

Picture?Picture?

Step 3: Do - Step 3: Do - Use the table.Use the table.

-3.6 ≤ z ≤-3.6 ≤ z ≤ 1.2 1.2 Area is: Area is: ((z ≤z ≤ 1.2) – ( 1.2) – (z ≤z ≤ -3.6) -3.6) Area isArea is: About 88.5%: About 88.5%

1.2-3.6

Step 4: Step 4: State your conclusion in context.State your conclusion in context.

About About 88.5%88.5% of all adult men are of all adult men are between 5 feet (60in.) and 6 feet tall.between 5 feet (60in.) and 6 feet tall.