Chapter 7 The Normal Probability Distribution

Chapter 7.1 Uniform and Normal DistributionObjective A: Uniform DistributionA1. Introduction

Recall: Discrete random variable probability distribution Special case: Binomial distribution

Finding the probability of obtaining success in independent trials of a binomial experiment is calculated by plugging the value of into the binomial formula as shown below:

Continuous Random variable

For a continued random variable the probability of observing one particular value is zero.

i.e.

Continuous Probability Distribution

We can only compute probability over an interval of values. Since and for a continuous variable,

To find probabilities for continuous random variables, we use probability density functions.

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Two common types of continuous random variable probability distribution: 1) Uniform distribution.

2) Normal distribution.

A2. Uniform distribution

Note: The area under a probability density function is 1.

for a uniform distribution

Example 1: A continuous random variable is uniformly distributed with . (a) Draw a graph of the uniform density function.

(b) What is ?

(c) What is ?

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ab 1

a b

430330 530 630 730X

1 1 2 2

Objective B: Normal distribution – Bell-shaped Curve

Example 1: Graph of a normal curve is given. Use the graph to identify the value of and .

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1 1 22 Z5.3 5.3

ZNegative ZPositive

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0 0

Example 2: The lives of refrigerator are normally distributed with mean years and

standard deviation years. (a) Draw a normal curve and the parameters labeled.

(b) Shade the region that represents the proportion of refrigerator that lasts for more than 17 years.

(c) Suppose the area under the normal curve to the right is . Provide two interpretations of this result.

Chapter 7.2 Applications of the Normal Distribution

Objective A: Area under the Standard Normal Distribution

The standard normal distribution – Bell shaped curve – =0 and =1

The random variable for the standard normal distribution is .

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Use the Z table (Table V) to find the area under the standard normal distribution. Each value in the body of the table is a cumulative area from the left up to a specific -score.Probability is the area under the curve over an interval.

The total area under the normal curve is 1.

Under the standard normal distribution,

(a) What is the area to the right ?

(b) What is the area to the left ?

Example 1: Draw the standard normal curve with the appropriate shaded area and use StatCrunch to determine the shaded area.

(a) that lies to the left of -1.38.

(b) that lies to the right of 0.56.

(c) that lies in between 1.85 and 2.47.

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5.0Area 5.0Area 5.0Area

Objective B: Finding the Z-score for a given probability

Example 1: Draw the standard normal curve and the Z-score such that the area to the left of the Z-score is 0.0418. Use StatCrunch to find the Z-score.

Example 2: Draw the standard normal curve and the Z-score such that the area to the right of the Z-score is 0.18. Use StatCrunch to find the Z-score.

Example 3: Draw the standard normal curve and two Z-scores such that the middle area of the standard normal curve is 0.70. Use StatCrunch to find the two Z-scores.

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Objective C: Probability under a Normal DistributionStep 1: Draw a normal curve and shade the desired area.

Step 2: Convert the values to -scores using .Step 3: Use StatCrunch to find the desired area.

Example 1: Assume that the random variable is normally distributed with mean

and a standard deviation .

(Note: this is not the standard normal curve because and .)

(a)

(b)

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Example 2: Redo Example 1 Use StatCrunch and random variable directly without converting to first.

(a)

(b)

Example 3: GE manufactures a decorative Crystal Clear 60-watt light bulb that it advertises will last 1,500 hours. Suppose that the lifetimes of the light bulbs are approximately

normal distributed, with a mean of 1,550 hours and a standard deviation of 57 hours, use StatCrunch to find what proportion of the light bulbs will last more than 1650 hours?

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Objective D: Finding the Value of a Normal Random Variable

Step 1: Draw a normal curve and shade the desired area. Step 2: Use StatCrunch to find the appropriate cutoff -score.

Step 3: Obtain from by the formula or .Example 1: The reading speed of 6th grade students is approximately normal (bell-shaped) with a mean speed of 125 words per minute and a standard deviation of 24 words per minute.

(a) What is the reading speed of a 6th grader whose reading speed is at the 90 percentile?

(b) Determine the reading rates of the middle 95% percentile.

Example 2: Redo Example 1 Use StatCruch to find directly without converting from to .(a) What is the reading speed of a 6th grader whose reading speed is at the 90 percentile?

(b) Determine the reading rates of the middle 95% percentile.

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Chapter 7.3 Normality Plot

Recall: A set of raw data is given, how would we know the data has a normal distribution? Use histogram or stem leaf plot. Histogram is designed for a large set of data.

For a very small set of data it is not feasible to use histogram to determine whether the data has a bell-shaped curve or not.

We will use the normal probability plot to determine whether the data were obtained from a normal distribution or not. If the data were obtained from a normal distribution, the data distribution shape is guaranteed to be approximately bell-shaped for n less than 30.

Perfect normal curve. The curve is aligned with the dots.

Almost a normal curve. The dots are within the boundaries.

Not a normal curve. Data is outside the boundaries.

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Example 1: Determine whether the normal probability plot indicates that the sample data could have come from a population that is normally distributed.

(a)

(b)

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