Normal Normal DistributionsDistributions

Density CurveDensity Curve

A density curve is a smooth function A density curve is a smooth function meant to approximate a histogram.meant to approximate a histogram.

The area under a density curve is The area under a density curve is one.one.

Density CurveDensity Curve

Density Curves: Density Curves: PropertiesProperties

Density CurvesDensity Curves

Mean of density curve is point at Mean of density curve is point at which the curve would balance.which the curve would balance.

For symmetric density curves, For symmetric density curves, balance point (mean) and the balance point (mean) and the median are the same.median are the same.

CharacterizationCharacterization A normal distribution A normal distribution

is bell-shaped and is bell-shaped and symmetric.symmetric.

The distribution is The distribution is determined by the determined by the mean mu, mean mu, and the and the standard deviation standard deviation sigma, sigma, ..

The mean mu The mean mu controls the center controls the center and sigma controls and sigma controls the spread.the spread.

DefinitionsDefinitions Mean is located in Mean is located in

center, or mode of center, or mode of normal curve.normal curve.

The standard The standard deviation is the deviation is the distance from the distance from the mean to the mean to the inflection point of the inflection point of the normal curve, the normal curve, the place where the place where the curve changes from curve changes from concave down to concave down to concave up.concave up.

ConstructionConstruction

A normal curve is drawn by first A normal curve is drawn by first drawing a normal curve.drawing a normal curve.

Next, place the mean, mu on the curve.Next, place the mean, mu on the curve. Then place sigma on curve by placing Then place sigma on curve by placing

the segment from the mean to the upper the segment from the mean to the upper (or lower) inflection point on your curve.(or lower) inflection point on your curve.

From this information, the scale on the From this information, the scale on the horizontal axis can be placed on the horizontal axis can be placed on the graph.graph.

ExamplesExamples

Draw normal curve with Draw normal curve with mean=mu=100, and standard mean=mu=100, and standard deviation = sigma = 10.deviation = sigma = 10.

Draw normal curve with mean = 20, Draw normal curve with mean = 20, sigma=2.sigma=2.

68-95-99.7 Rule68-95-99.7 Rule For any normal curve For any normal curve

with mean mu and with mean mu and standard deviation sigma:standard deviation sigma:

68 percent of the 68 percent of the observations fall within observations fall within one standard deviation one standard deviation sigma of the mean.sigma of the mean.

95 percent of observation 95 percent of observation fall within 2 standard fall within 2 standard deviations.deviations.

99.7 percent of 99.7 percent of observations fall within 3 observations fall within 3 standard deviations of standard deviations of the mean.the mean.

Example QuestionsExample Questions

If mu=30 and sigma=4, what are the If mu=30 and sigma=4, what are the values (a, b) around 30 such that 95 values (a, b) around 30 such that 95 percent of the observations fall percent of the observations fall between these values?between these values?

If mu=40 and sigma=5, what are the If mu=40 and sigma=5, what are the bounds (a, b) such that 99.7 percent of bounds (a, b) such that 99.7 percent of the values fall between these values?the values fall between these values?

Standard Normal Standard Normal DistributionDistribution

The standard normal distribution has mean The standard normal distribution has mean = 0 and standard deviation sigma=1.= 0 and standard deviation sigma=1.

Normal Table UsageNormal Table Usage What proportion of standard normal distribution What proportion of standard normal distribution

values Z are less than 1.40? That is, P(Z < 1.40) values Z are less than 1.40? That is, P(Z < 1.40) = ?= ?

Ans:.9192 or 91.92 percent of values.Ans:.9192 or 91.92 percent of values.

Standard NormalStandard Normal

P( 0 < Z < 1.40) = ?P( 0 < Z < 1.40) = ? Ans: P(Z < 1.40) – P(Z<0) = .9192 Ans: P(Z < 1.40) – P(Z<0) = .9192

- .5 = .4192- .5 = .4192

ExampleExample

P( Z < - 2.15) = ?P( Z < - 2.15) = ?

Normal Table UsageNormal Table Usage P( .64 < Z < 1.23) = ?P( .64 < Z < 1.23) = ? Ans: P(Z<1.23) – P(Z < .64) = .8907 - .7389 Ans: P(Z<1.23) – P(Z < .64) = .8907 - .7389

= .1518= .1518 P(Z > 2.24) = CAREFUL !!!!! P(Z > 2.24) = CAREFUL !!!!! Ans: Either = 1 – P(Z < 2.24) = 1 - .9875 =Ans: Either = 1 – P(Z < 2.24) = 1 - .9875 = or by symmetry = P(Z < - 2.24) = .0125. In or by symmetry = P(Z < - 2.24) = .0125. In

this approach you are using the fact that this approach you are using the fact that both tails of a standard normal are the same both tails of a standard normal are the same and so P(Z>2.24) = P(Z< -2.24) = .0125.and so P(Z>2.24) = P(Z< -2.24) = .0125.

Z-Score FormulaZ-Score Formula Any normal distribution with mean=mu and Any normal distribution with mean=mu and

standard deviation= sigma, can be converted standard deviation= sigma, can be converted into a standard normal Z distribution by the into a standard normal Z distribution by the following transformation:following transformation:

ExampleExample Consider a distribution with mean=mu=100 Consider a distribution with mean=mu=100

and standard deviation = sigma = 10. Draw and standard deviation = sigma = 10. Draw density curve with number line provided.density curve with number line provided.

Now re-draw the curve and number line on Now re-draw the curve and number line on horizontal axis after subtracting 100 from horizontal axis after subtracting 100 from each value. Notice this centers the curve at each value. Notice this centers the curve at zero.zero.

Then draw the resulting number line after Then draw the resulting number line after dividing the previous number line values by dividing the previous number line values by 10.10.

Voila ! We are now back to Z scale !Voila ! We are now back to Z scale !

ExampleExample

Example 1.26Example 1.26 in Page 75. in Page 75. X=The X=The SATSAT score of a randomly score of a randomly

chosen student. X has Nchosen student. X has N1019, 1019, =209).=209).

What percent of all students had What percent of all students had SAT scores of at least 820? That is, SAT scores of at least 820? That is, P( X > 820) = ?P( X > 820) = ?

SolutionSolution

P( X > 820 ) = P( X > 820 ) =

Solution = .8289Solution = .8289

820X 820X 820X 820X

ExampleExample

Problem 1.86 Problem 1.86 (Moore&Mc)(Moore&Mc)

Eleanor gets 680 Eleanor gets 680 on SAT math exam. on SAT math exam. Mean on this Mean on this exam is 500 and sd exam is 500 and sd is 100.is 100.

Eleanor’s Eleanor’s standardized score standardized score is:is:

1.86 Continued1.86 Continued

Gerald got 27 on Gerald got 27 on ACT math. Mean is ACT math. Mean is 18 with sd of 6.18 with sd of 6.

Gerald’s Z-Score is:Gerald’s Z-Score is:

Eleanor did better !Eleanor did better !

Human PregnanciesHuman Pregnancies

What proportion of births are premature? What proportion of births are premature? That is, what proportion is below 240 days? That is, what proportion is below 240 days? P(X<240)= ?P(X<240)= ?

London Bus DriversLondon Bus Drivers

Calorie intake for drivers averages 2821 cals Calorie intake for drivers averages 2821 cals per day with sd=sigma=436.per day with sd=sigma=436.

What proportion of drivers have calorie intakes, What proportion of drivers have calorie intakes, X, less than 2000 calories per day? P(X < 2000)?X, less than 2000 calories per day? P(X < 2000)?

0301.)88.1(

)436

28212000()2000(

ZP

XPXP

London Bus DriversLondon Bus Drivers What proportion of drivers consume between What proportion of drivers consume between

2000 and 2500 cals per day? P(2000<X<2500)?2000 and 2500 cals per day? P(2000<X<2500)?

1995.0301.2296.)74.88.1(

)436

28212500

436

28212000()25002000(

ZP

XPXP

Finding a PercentileFinding a Percentile

Backwards problem. We are now given a Backwards problem. We are now given a fraction and need to find the X-value. fraction and need to find the X-value.

In past, we were provided X and found a In past, we were provided X and found a proportion.proportion.

Use Formula:Use Formula:

*ZX

London Bus DriversLondon Bus Drivers

Find the calorie intake at the 90Find the calorie intake at the 90thth percentile of the calorie distribution.percentile of the calorie distribution.

Insert mean and sd into backward formula, Insert mean and sd into backward formula, then determine correct Z-star value.then determine correct Z-star value.

08.3379)436(28.12821

)436(2821 **

X

ZZX

Finding a PercentileFinding a Percentile Plugging in the mean and sd are not hard. The Plugging in the mean and sd are not hard. The

difficulty is finding Z-star. It is simply the same difficulty is finding Z-star. It is simply the same percentile you are trying to find, except for the percentile you are trying to find, except for the standard normal distribution. This requires you to standard normal distribution. This requires you to use an inverse lookup in your z-table.use an inverse lookup in your z-table.

*ZX

TV ViewingTV Viewing Neilsen ratings service found that tv viewing Neilsen ratings service found that tv viewing

for children aged 2-11 had a normal for children aged 2-11 had a normal distribution with mean 23.02 hours and distribution with mean 23.02 hours and sigma=6.23 hours.sigma=6.23 hours.

What proportion of children watch more than What proportion of children watch more than 24 hours of tv per week?24 hours of tv per week?

4364.)16.(

)23.6

02.2324()24(

ZP

XPXP

TV ViewingTV Viewing How many hours of tv does a child watch that is at How many hours of tv does a child watch that is at

the 95the 95thth percentile of the tv viewing distribution? percentile of the tv viewing distribution?

24.33)23.6(64.102.23* ZX

The Central Limit The Central Limit TheoremTheorem

(for the sample mean x)(for the sample mean x) If a random sample of n observations If a random sample of n observations

is selected from a population (is selected from a population (anyany population), then when n is population), then when n is sufficiently large, sufficiently large, the sampling the sampling distribution of x will be distribution of x will be approximately normal.approximately normal.

(The larger the sample size, the better (The larger the sample size, the better will be the normal approximation to will be the normal approximation to the sampling distribution of x.)the sampling distribution of x.)

How Large Should n Be? How Large Should n Be?

For the purpose of applying the For the purpose of applying the central limit theorem, we will central limit theorem, we will consider a sample size to be large consider a sample size to be large when n > 30.when n > 30.