Objectives

1. Use the uniform probability distribution

2. Graph a normal curve

3. State the properties of the normal curve

4. Explain the role of area in the normal density function

Objective – Use the uniform probability distribution

- – a distribution in which all intervals of

the same length are equally likely to occur

- – an equation used to compute

probabilities of continuous random variables. Must satisfy the following two properties:

o The total area under the graph of the equation must equal 1

o The height of the graph of the equation must be greater than or equal to 0 for all possible

values of the random variable

Math 203 Notes – Chapter 7 - The Normal Probability Distribution

Section 7.1 – The Standard Normal Curve & Properties of the Normal Distribution

Example: Illustrating the Uniform Distribution

Suppose that United Parcel Service is supposed to deliver a package to your front door and the arrival time

is somewhere between 10 am and 11 am. Let the random variable x represent the time from 10 am when

the delivery is supposed to take place. The delivery could be at 10 am (x = 0) or at 11 am (x = 60) with all

1-minute interval of times between x = 0 and x = 60 equally likely.

That is, your package is just as likely to arrive between 10:15 and 10:16 as it is to arrive between 10:40 and

10:41. The random variable x can be any value in the interval from 0 to 60, that is, 0 < x < 60.

Because any two intervals of equal length between 0 and 60, inclusive, are equally likely, the random

variable x is said to follow a uniform probability distribution.

The graph illustrates the properties for the example. Notice the area of

the rectangle is and the graph is greater than or equal to zero

for all x between 0 and 60, inclusive.

Because the area of a rectangle is height⋅width, and the width of the

rectangle is 60, the height must be in order for the

area to be 1.

Values of the random variable x less than 0 or greater than 60 are

, thus the function value must

be zero for x less than 0 or greater than 60.

Big Concept:

The area under the graph of a density function over an interval represents the

of observing a value of the random variable in that interval.

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Example: Area as a Probability

Using the information from the UPS example, answer the following:

1) What is the probability that UPS delivers the package between 10:15 and 10:30 am (inclusive)?

Analysis: The probability of a delivery time that is between 15 and 30 minutes after the hour is the

area under the uniform density function.

2) What is the probability that UPS delivers the package between 10:50 and 11 am (inclusive)?

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Objective – Graph a Normal Curve

- Relative frequency histograms that are symmetric and bell-shaped are said to have the shape of a

- If a continuous random variable is normally distributed

or has a normal probability distribution, then a relative

frequency histogram of the random variable has the shape

of a normal curve – bell-shaped and symmetric

o This also means the

can be applied

Objective – State the Properties of the Normal Curve

Properties of the Normal Density Curve

1. It is symmetric about its mean, µ

2. Because mean = median = mode, the curve has a

single peak and the highest point occurs at x = µ

3. It has inflection points at µ – σ and µ + σ

4. The under the curve is

5. The area under the curve to the right of µ is

to the area under the curve to

the left of µ, which equals 1/2.

6. As x increases to infinity or decreases to negative

infinity, the graph approaches, but never reaches, the

horizontal axis.

7. The Empirical Rule applies

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Objective – Explain the Role of Area in the Normal Density Function

- – if a random variable x is normally

distributed with mean µ and standard deviation σ, the under the normal curve

for any interval of values of the random variable x represents either:

o The of the population with the characteristic

described by the interval of values

o The that a randomly selected individual from the

population will have the characteristic described by the interval of values

Example: Interpreting the Area Under a Normal Curve

The weights of giraffes are approximately normally distributed with mean µ = 2200 pounds and standard

deviation σ = 200 pounds.

a) Draw a normal curve with the parameters labeled and shade the area under the normal curve to the left

of x = 2100 pounds.

b) Suppose that the area under the normal curve to the left of x = 2100 pounds is 0.3085. Provide two

interpretations of this result.

The of giraffes whose weight is less than 2100 pounds is

The that a randomly selected giraffe weighs less than 2100 pounds

is

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Section 7.2 – Applications of the Normal Distribution

Objectives

1. Find and interpret the area under a normal curve

2. Find the value of a normal random variable

Objective – Find and interpret the area under a normal curve

We are now going to standardize the Normal Curve:

The Normal Curve The Standard Normal Curve

Standardizing a Normal Random Variable

Suppose that the random variable x is normally distributed with mean µ and standard deviation σ. Then the

random variable

! = ! − !!

is normally distributed with mean and standard deviation

The random variable z is said to have the standard normal distribution.

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Example: Reviewing z-scores

IQ scores can be modeled by a normal distribution with µ = 100 and σ = 15. Find how many standard

deviations above the mean is an individual whose IQ score is 120.

What if we are asked what proportion of individuals has an IQ score of 120 or higher?

We would need to find the under the standard normal curve.

To find the Area under the Standard Normal Curve given z-scores:

On your calculator, we use the function.

To find the function, hit:

2nd, Dist (vars), arrow to normalcdf, Enter

We then input the following details corresponding to the given z-scores:

normalcdf (lower limit, upper limit)

*Note: The TI-84 does not have a button for infinity, so we use +10^99 to represent infinity when

needed

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Examples: Finding the Area Under the Standard Normal Curve

Find the area under the standard normal curve to the left of the following z-scores and shade the corresponding area under the provided curve. Also show what you input into your calculator. 1) z = -1.38 2) z = 0.87

Find the area under the standard normal curve to the right of the following z-scores and shade the

corresponding area under the provided curve. Also show what you input into your calculator.

3) z = -0.21 4) z = 1.25

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Find the area under the standard normal curve between the following z-scores and shade the corresponding

area under the provided curve. Also show what you input into your calculator.

5) z = -1.21 and 0.98 6) z = 0.33 and 2.25

To find the Area under any Normal Curve:

Step 1: Draw a normal curve and shade the desired area.

Step 2: Convert the value(s) of x to z-score(s) using ! = !!!!

Step 3: Draw a standard normal curve and shade the desired area.

Step 4: Use the normalcdf calculator function to find the area/probability/percentage for the z-score(s)

found in Step 2.

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Examples: Finding the Area Under Any Normal Curve

1) IQ scores can be modeled by a normal distribution with µ = 100 and σ = 15.

a. Find the proportion of individuals that have an IQ score of 120 or higher?

b. What is the probability that a randomly selected individual has an IQ between 80 and 125?

normalcdf(120,10^99,100,15) OR

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c. Find the percentile ranking of an individual whose IQ is 132.

2) In sports betting, Las Vegas sports books establish a “spread” – winning margins for a team that is favored to win a game. In a certain category of games, the margin of victory for the favored team relative to the spread is approximately normally distributed with a mean of 0 points and a standard deviation of 10.9 points. Source: Justin Wolfers, “Point Shaving: Corruption in NCAA Basketball”

a. What is the probability that the favored team wins by 5 or more points relative to the spread?

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b. What is the probability that the favored team loses by 2 or more points relative to the spread?

c. What proportion of games result in the favored team either winning or losing by 12 or more points

relative to the spread?

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Objective – Find the Value of a Normal Random Variable

Finding the Value of a Normal Random Variable Corresponding to a Specified Proportion, Probability, or Percentile Step 1: Draw a normal curve and shade the area corresponding to the proportion, probability, or percentile. Step 2: Use the invNorm calculator function to find the z-score that corresponds to the shaded area. Step 3: Obtain the normal value from the formula

! = ! + !"

On your calculator, to use the invNorm function, hit: 2nd, Dist (vars), arrow to invNorm, Enter

Then input: invNorm(area to the left of the desired z-score)

Example: Finding the z-score of a Normal Random Variable

Find the z-score that corresponds to the 85th percentile.

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Example: Finding the Value of a Normal Random Variable in Context of a Problem

Scores on the GRE are normally distributed with mean 1049 and standard deviation 189. (Source:

http://www.ets.org/Media/Tests/GRE/pdf/994994.pdf.) What is the score (on the test, not z-score) of a student

whose percentile rank is at the 90th percentile?

Interpretation: A person who scores on the GRE would rank in the 90th percentile.

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Example: Finding the Value of a Normal Random Variable in Context of a Problem

It is known that the length of a certain steel rod is normally distributed with a mean of 100 cm and a

standard deviation of 0.45 cm. Suppose the manufacturer wants to accept 90% of all rods manufactured.

Determine the length of rods that make up the middle 90% of all steel rods manufactured.

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Follow-up concept: !! definition

The notation

(pronounced “z sub alpha”) is the z-score such that the area under

the standard normal curve to the of !! is !.

Examples: Finding the values of !!

Find the value of !! for the following questions.

1) !!.!" 2) !!.!"

Importantvocabulary!

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Objective – Use normal probability to assess normality

Suppose that we obtain a simple random sample from a population whose distribution is

Many of the statistical tests that we will perform on small data sets (sample size less than 30) require that

the population from which the sample is drawn be

Up to this point, we have said that a random variable x is normally distributed, or at least approximately

normal, provided the histogram of the data is symmetric and bell-shaped.

This method works well for large data sets, but the shape of a histogram drawn from a small sample of

observations does not always accurately represent the shape of the population.

*Main point: When sample size is small, we need additional methods for assessing the

of a random variable x when we are looking at sample data. To do this, we will use a normal

probability plot.

- – the z-score of the data value

the distribution of the random variable were to be normal

- – graphs observed data on the x-axis

and the normal z-scores on the y-axis

o If sample data is taken from a population that is normally distributed, a normal probability

plot will be approximately (remember this!)

o It is difficult to determine whether a normal probability plot is “linear enough.” If the linear correlation coefficient (Ch.4) between the expected z-scores and the observed data is greater than the critical value for the sample size, then it is reasonable to conclude that the data could come from a population that is normally distributed.

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Section 7.6 – Assessing Normality

Discussion Example: Interpreting a Normal Probability Plot

A random sample of weekly work logs at an automobile repair station was obtained, and the average

number of customers served per day was recorded. The data is given below. Use the provided normal

probability plot to assess whether the sample data could have come from a population that is normally

distributed.

26 24 22 25 23 24 25 23 25 22 21 26 24 23 24 25 24 25 24 25 26 21 22 24 24

Normal Probability Plot of the Sample Data

Conclusion: The normal probability plot of the data Therefore, we conclude that the sample data come from a

population that is normally distributed.

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Example: Constructing a Normal Probability Plot

A random sample of 20 undergraduate students receiving student loans was obtained, and the amount of

their loans (in dollars) for the 2011–2012 school year was recorded. The data is given below. Determine

whether the data could have come from a population that is normally distributed.

2,500 1,000 2,000 14,000 1,800 3,800 10,100 2,200 900 1,600 500 2,200 6,200 9,100 2,800

2,500 1,400 13,200 750 12,000

On your calculator:

Input the data in any list (I’m using L3).

Hit 2nd, Stat Plot, Enter (to enter Plot1 details), highlight On, Enter

Then fill in the following details:

Type: arrow to the last figure (that’s the normal Probability Plot)

Data List: L3 (or whichever list you used)

Data Axis: X

Mark: Any will work

To generate the graph: Hit Zoom, arrow to ZoomStat (option 9), Enter Conclusion:

The normal probability plot does not appear to be linear. Therefore the sample data is from a population that is likely not normally distributed.

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Discussion Example: Interpreting a Normal Probability Plot

Suppose that seventeen randomly selected workers at a detergent factory

were tested for exposure to a Bacillus subtillis enzyme by measuring the

ratio of forced expiratory volume (FEV) to vital capacity (VC). A normal

probability plot of the sample data is provided below. Is it reasonable to

conclude that the FEV to VC (FEV/VC) ratio is normally distributed?

Source: Shore, N.S.; Greene R.; and Kazemi, H. “Lung Dysfunction in Workers Exposed to

Bacillus subtillis Enzyme,” Environmental Research, 4 (1971), pp. 512 - 519.

In case you’re interested: FEV is the maximum volume of air a person can exhale in one

second; VC is the maximum volume of air that a person can exhale after taking a deep

breath.

Normal Probability Plot of the Sample Data

0.61 0.7

0.84 0.63

0.78 0.85

0.73 0.82

0.67 0.74

0.87 0.88

0.76 0.85

0.72 0.83

0.64

The normal probability plot does appear to be linear. Therefore the sample data is from a population that is likely normally distributed.

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