sst5e_ppt_ch07.pptx

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STATISTICS

INFORMED DECISIONS USING DATA

Fifth Edition, Global Edition

Chapter 7

The Normal

Probability

Distribution

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7.1 Properties of the Normal Distribution

Learning Objectives

1. Use the uniform probability distribution

2. Graph a normal curve

3. State the properties of the normal curve

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7.1 Properties of the Normal Distribution

7.1.1 Use the Uniform Probability Distribution

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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 to say 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

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7.1 Properties of the Normal Distribution

7.1.1 Use the Uniform Probability Distribution

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A

probability density function (pdf)

is an equation used to

compute probabilities of continuous random variables. It

must satisfy the following two properties:

1. The total area under the graph of the equation over all possible values of the random variable must equal 1.

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7.1 Properties of the Normal Distribution

7.1.1 Use the Uniform Probability Distribution

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7.1 Properties of the Normal Distribution

7.1.1 Use the Uniform Probability Distribution

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7.1 Properties of the Normal Distribution

7.1.1 Use the Uniform Probability Distribution

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The area under the graph of the density function over an

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7.1 Properties of the Normal Distribution

7.1.1 Use the Uniform Probability Distribution

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EXAMPLE Area as a Probability

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7.1 Properties of the Normal Distribution

7.1.2 Graph a Normal Curve

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7.1 Properties of the Normal Distribution

7.1.2 Graph a Normal Curve

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A continuous random variable is normally distributed, or has a normal probability distribution, if its relative frequency

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7.1 Properties of the Normal Distribution

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7.1 Properties of the Normal Distribution

7.1.3 State the Properties of the Normal Curve

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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

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7.1 Properties of the Normal Distribution

7.1.3 State the Properties of the Normal Curve

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6. As x increases without bound (gets larger and larger), the

graph approaches, but never reaches, the horizontal axis. As x decreases without bound (gets more and more negative), the graph approaches, but never reaches, the horizontal axis.

7. The Empirical Rule: Approximately 68% of the area under the normal curve is between x = μ − σ and x = μ + σ; approximately 95% of the area is between x = μ − 2σ and x = μ + 2σ;

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7.1 Properties of the Normal Distribution

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7.1 Properties of the Normal Distribution

7.1.4 Explain the Role of Area in the Normal Density Function (1 of 9)

EXAMPLE A Normal Random Variable

The data on the next slide represent the heights (in inches) of a random sample of 50 two-year old males.

a) Draw a histogram of the data using a lower class limit of the first class equal to 31.5 and a class width of 1.

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7.1 Properties of the Normal Distribution

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7.1 Properties of the Normal Distribution

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7.1 Properties of the Normal Distribution

In the next slide, we have a normal density curve drawn

over the histogram. How does the area of the rectangle

corresponding to a height between 34.5 and 35.5 inches

relate to the area under the curve between these two

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7.1 Properties of the Normal Distribution

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7.1 Properties of the Normal Distribution

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7.1 Properties of the Normal Distribution

Area under a Normal Curve

Suppose that a random variable X is normally distributed with mean μ and standard deviation σ. The area under the normal curve for any interval of values of the random variable X

represents either

• the proportion of the population with the characteristic described by the interval of values or

• the probability that a randomly selected individual from the

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7.1 Properties of the Normal Distribution

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.

(b) Shade the area under the normal curve to the left of x = 2100 pounds.

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7.1 Properties of the Normal Distribution

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• The proportion of giraffes whose weight is less than 2100 pounds is 0.3085

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7.2 Applications of the Normal Distribution

Learning Objectives

1. Find and interpret the area under a normal curve

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7.2 Applications of the Normal Distribution

7.2.1 Find and Interpret the Area Under a Normal Curve (1 of 14)

Standardizing a Normal Random Variable

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7.2 Applications of the Normal Distribution

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7.2 Applications of the Normal Distribution

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7.2 Applications of the Normal Distribution

IQ scores can be modeled by a normal distribution with

μ

= 100 and

σ

= 15.

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7.2 Applications of the Normal Distribution

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7.2 Applications of the Normal Distribution

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7.2 Applications of the Normal Distribution

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7.2 Applications of the Normal Distribution

EXAMPLE Finding the Area Under the Standard Normal Curve

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7.2 Applications of the Normal Distribution

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7.2 Applications of the Normal Distribution

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7.2 Applications of the Normal Distribution

Find the area under the standard normal curve between z = −1.02 and z = 2.94.

Area between

−

1.02 and 2.94

= (Area left of z = 2.94) − (area left of z = −1.02)

= 0.9984 − 0.1539

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7.2 Applications of the Normal Distribution

Solution:

• Convert the value of x to a z-score. Use Table V to find the row and column that correspond to z. The area to the left of x is the value where the row and column intersect.

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7.2 Applications of the Normal Distribution

Solution:

• Convert the value of x to a z-score. Use Table V to find the area to the left of z (also is the area to the left of x). The area to the right of

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7.2 Applications of the Normal Distribution

Solution:

• Convert the values of x to a z-scores. Use Table V to find the area to the left of z₁ and to the left of z₂. The area between z₁ and z₂ is (area to the left of z₂) − (area to the left of z₁).

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7.2 Applications of the Normal Distribution

7.2.2 Find the Value of a Normal Random Variable

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Procedure for Finding the Value of a Normal Random Variable

Step 1: Draw a normal curve and shade the area corresponding to the proportion, probability, or percentile.

Step 2: Use Table V to find the z-score that corresponds to the shaded area.

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7.2 Applications of the Normal Distribution

7.2.2 Find the Value of a Normal Random Variable

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EXAMPLE Finding the Value of a Normal Random Variable

The combined (verbal + quantitative reasoning) score on the GRE is normally distributed with mean 1049 and standard deviation 189.

(Source: http://www.ets.org/Media/Tests/GRE/pdf/994994.pdf.)

What is the score of a student whose percentile rank is at the 85th

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7.2 Applications of the Normal Distribution

7.2.2 Find the Value of a Normal Random Variable

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The z-score that corresponds to the 85th percentile is the z-score

such that the area under the standard normal curve to the left is 0.85. This z-score is 1.04.

x = µ + zσ

= 1049 + 1.04(189)

= 1246

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7.2 Applications of the Normal Distribution

7.2.2 Find the Value of a Normal Random Variable

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It is known that the length of a certain steel rod is normally

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7.2 Applications of the Normal Distribution

7.2.2 Find the Value of a Normal Random Variable

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7.2 Applications of the Normal Distribution

7.2.2 Find the Value of a Normal Random Variable

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