TUTORIAL 8 - Normal Distributions

February 12, 2018 | Author: Anonymous | Category: Math, Statistics And Probability, Normal Distribution
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Statistics Workshop Tutorial 8 •

Normal Distributions

Slide 2

Section 5-1 Overview Created by Erin Hodgess, Houston, Texas

Copyright © 2004 Pearson Education, Inc.

LAPTOP3:

Overview

Slide 3

 Continuous random variable

 Normal distribution -1

x- ( ) 2

f(x) =

e



2

2p

Formula 5-1 Figure 5-1 Copyright © 2004 Pearson Education, Inc.

Slide 4

Section 5-2 The Standard Normal Distribution

Copyright © 2004 Pearson Education, Inc.

Definitions  Uniform Distribution

Slide 5

is a probability

distribution in which the continuous random variable values are spread evenly over the range of possibilities; the graph results in a rectangular shape.

Copyright © 2004 Pearson Education, Inc.

Definitions

Slide 6

 Density Curve (or probability density function is the graph of a continuous probability distribution.

1. The total area under the curve must equal 1. 2. Every point on the curve must have a vertical height that is 0 or greater. Copyright © 2004 Pearson Education, Inc.

Slide 7

Because the total area under the density curve is equal to 1, there is a correspondence between area and probability.

Copyright © 2004 Pearson Education, Inc.

Using Area to Find Probability

Figure 5-3 Copyright © 2004 Pearson Education, Inc.

Slide 8

Heights of Adult Men and Women Slide 9

Figure 5-4 Copyright © 2004 Pearson Education, Inc.

Definition

Slide 10

Standard Normal Distribution: a normal probability distribution that has a mean of 0 and a standard deviation of 1.

Figure 5-5 Copyright © 2004 Pearson Education, Inc.

Slide 11

Example:

If thermometers have an average (mean) reading of 0 degrees and a standard deviation of 1 degree for freezing water and if one thermometer is randomly selected, find the probability that, at the freezing point of water, the reading is less than 1.58 degrees.

P (z < 1.58) = 0.9429

The probability that the chosen thermometer will measure freezing water less than 1.58 degrees is 0.9429. Copyright © 2004 Pearson Education, Inc.

Slide 12

Copyright © 2004 Pearson Education, Inc.

Slide 13

Copyright © 2004 Pearson Education, Inc.

Finding z Scores when Given Probabilities

Slide 14

5% or 0.05

(z score will be positive)

1.645

Figure 5-10 Finding the 95th Percentile Copyright © 2004 Pearson Education, Inc.

Slide 15

Section 5-3 Applications of Normal Distributions Created by Erin Hodgess, Houston, Texas

Copyright © 2004 Pearson Education, Inc.

Nonstandard Normal Distributions





Slide 16

If  0 or  1 (or both), we will convert values to standard scores using Formula 5-2, then procedures for working with all normal distributions are the same as those for the standard normal distribution.

Formula 5-2

z=

x–µ



Copyright © 2004 Pearson Education, Inc.

Converting to Standard Normal Distribution

z=

x– 

Figure 5-12

Copyright © 2004 Pearson Education, Inc.

Slide 17

Probability of Weight between Slide 18 140 pounds and 211 pounds In the Chapter Problem, we noted that the Air Force had been using the ACES-II ejection seats designed for men weighing between 140 lb and 211 lb. Given that women’s weights are normally distributed with a mean of 143 lb and a standard deviation of 29 lb (based on data from the National Health survey), what percentage of women have weights that are within those limits?

Copyright © 2004 Pearson Education, Inc.

Probability of Weight between Slide 19 140 pounds and 211 pounds  = 143  = 29

There is a 0.5302 probability of randomly selecting a woman with a weight between 140 and 211 lbs. (53.02% of women)

Figure 5-14 Copyright © 2004 Pearson Education, Inc.

Find P98 for Hip Breadths of Men x =  + (z ● ) x = 14.4 + (2.05  1.0) x = 16.45

Slide 20

The hip breadth of 16.5 in. separates the lowest 98% from the highest 2%

Seats designed for a hip breadth up to 16.5 in. will fit 98% of men.

Figure 5-15 Copyright © 2004 Pearson Education, Inc.

Slide 21

Section 5-4 Sampling Distributions and Estimators Created by Erin Hodgess, Houston, Texas

Copyright © 2004 Pearson Education, Inc.

Definition

Slide 22

Sampling Distribution of the mean is the probability distribution of sample means, with all samples having the same sample size n.

Copyright © 2004 Pearson Education, Inc.

Definitio n Sampling Variability:

Slide 23

The value of a statistic, such as the sample mean x, depends on the particular values included in the sample.

Copyright © 2004 Pearson Education, Inc.

Slide 24

Section 5-5 The Central Limit Theorem Created by Erin Hodgess, Houston, Texas

Copyright © 2004 Pearson Education, Inc.

Central Limit Theorem

Slide 25

Given: 1. The random variable x has a distribution (which may or may not be normal) with mean µ and standard deviation . 2. Samples all of the same size n are randomly selected from the population of x values.

Copyright © 2004 Pearson Education, Inc.

Central Limit Theorem

Slide 26

Conclusions: 1. The distribution of sample x will, as the sample size increases, approach a normal distribution. 2. The mean of the sample means will be the population mean µ. 3. The standard deviation of the sample means n will approach 

Copyright © 2004 Pearson Education, Inc.

Practical Rules Commonly Used:

Slide 27

1. For samples of size n larger than 30, the distribution of the sample means can be approximated reasonably well by a normal distribution. The approximation gets better as the sample size n becomes larger. 2. If the original population is itself normally distributed, then the sample means will be normally distributed for any sample size n (not just the values of n larger than 30).

Copyright © 2004 Pearson Education, Inc.

Notation

Slide 28

the mean of the sample means

µx = µ the standard deviation of sample mean

 x = n (often called standard error of the mean) Copyright © 2004 Pearson Education, Inc.

Slide 29

Section 5-6 Normal as Approximation to Binomial Created by Erin Hodgess, Houston, Texas

Copyright © 2004 Pearson Education, Inc.

Approximate a Binomial Distribution Slide 30 with a Normal Distribution if: np  5 nq  5 then µ = np and  =

npq

and the random variable has

a

distribution. (normal) Copyright © 2004 Pearson Education, Inc.

Finding the Probability of Slide 31 “At Least” 120 Men Among 200 Accepted Applicants

Figure 5-24 Copyright © 2004 Pearson Education, Inc.

Definition

Slide 32

When we use the normal distribution (which is continuous) as an approximation to the binomial distribution (which is discrete), a continuity correction is made to a discrete whole number x in the binomial distribution by representing the single value x by the interval from x – 0.5 to x + 0.5. Copyright © 2004 Pearson Education, Inc.

x = at least 120 = 120, 121, 122, . . .

x = more than 120 = 121, 122, 123, . . .

x = at most 120 = 0, 1, . . . 118, 119, 120

x = fewer than 120 = 0, 1, . . . 118, 119 Figure 5-25 Copyright © 2004 Pearson Education, Inc.

Slide 33

x = exactly 120

Interval represents discrete number 120 Copyright © 2004 Pearson Education, Inc.

Slide 34

Slide 35

Section 5-7 Determining Normality Created by Erin Hodgess, Houston, Texas

Copyright © 2004 Pearson Education, Inc.

Definition

Slide 36

A Normal Quantile Plot is a graph of points (x,y), where each x value is from the original set of sample data, and each y value is a z score corresponding to a quantile value of the standard normal distribution.

Copyright © 2004 Pearson Education, Inc.

Procedure for Determining Whether Data Have a Normal Distribution

Slide 37

1. Histogram: Construct a histogram. Reject normality if the histogram departs dramatically from a bell shape. 2. Outliers: Identify outliers. Reject normality if there is more than one outlier present. 3. Normal Quantile Plot: If the histogram is basically symmetric and there is at most one outlier, construct a normal quantile plot as follows:

Copyright © 2004 Pearson Education, Inc.

Procedure for Determining Whether Data Have a Normal Distribution

Slide 38

Examine the normal quantile plot and reject normality if the the points do not lie close to a straight line, or if the points exhibit some systematic pattern that is not a straight-line pattern.

Copyright © 2004 Pearson Education, Inc.

Now we are ready for Day 3

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