The Normal Probability Distribution

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The Normal Probability Distribution

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The Normal Probability Distribution. What is a distribution? A collection of scores, values, arranged to indicate how common various values, or scores are. Mean (population, sample) Standard deviation (population, sample) Median Mode. Scores in our class. - PowerPoint PPT Presentation

Transcript of The Normal Probability Distribution

Page 1: The Normal Probability                       Distribution

The Normal Probability

Distribution

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What is a distribution? A collection of scores, values, arranged to indicate how common various values, or scores are.

Mean (population, sample)Standard deviation (population, sample)MedianMode

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Scores in our class

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CHARACTERISTICS OF A NORMAL DISTRIBUTIONCHARACTERISTICS OF A NORMAL DISTRIBUTION

Theoretically, curveextends to - infinity

Theoretically, curve extends to + infinityMean, median, and

mode are equal

Tail Tail

Normal curve is symmetrical - two halves identical -

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AREAS UNDER THE NORMAL CURVEAREAS UNDER THE NORMAL CURVE

About 68 percent of the area under the normal curve is within plus one and minus one standard deviation of the mean. This can be written as ± 1.

About 95 percent of the area under the normal curve is within plus and minus two standard deviations of the mean, written ± 2.

Practically all (99.74 percent) of the area under the normal curve is within three standard deviations of the mean, written ± 3.

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Between:

68.26%

95.44%

99.97%

Between:

68.26%

95.44%

99.97%

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Normal Distributions with Equal Means but Different Standard Deviations.

Normal Distributions with Equal Means but Different Standard Deviations.

3.9 = 5.0

3.9 = 5.0

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Normal Probability Distributions with Different Means and Standard Deviations.

Normal Probability Distributions with Different Means and Standard Deviations.

= 5, = 3 = 9, = 6 = 14, = 10

= 5, = 3 = 9, = 6 = 14, = 10

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What is this good for??

• describes the data and how it clusters, arranges around a mean.• it’s good for us because it can allow us to make statistical inferences

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CHARACTERISTICS OF A NORMAL CHARACTERISTICS OF A NORMAL PROBABILITY DISTRIBUTIONPROBABILITY DISTRIBUTION

A normal distribution with a mean of 00 and a standard deviation of 11 is called the standard normal standard normal distributiondistribution.

z value:z value: The distance between a selected value, designated X, and the population mean , divided by the population standard deviation, .

XZ XZ

Disguised under z-score, normal scores, standardized score

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What is it good for?Indicates how many standard deviations an

observation is above/below the meanIt’s good, because it allows us to compare

observations from other normal distributions

Is a 3.00 GPA UNLV student as good as a 3.00 GPA UCF student?

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EXAMPLE 1EXAMPLE 1

The monthly incomes of recent high school graduates in a large corporation are normally distributed with a mean of $2,000 and a standard deviation of $200. What is the z value for an income X of $2,200? $1,700?

For X = $2,200 and since z = (X - then z =.

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EXAMPLE 1 (continued)EXAMPLE 1 (continued)

For X = $1,700 and since z = (X - then

A z value of +1.0 indicates that the value of $2,200 is ___ standard deviation ______ the mean of $2,000.

A z value of – 1.5 indicates that the value of $1,700 is ____ standard deviation ______ the mean of $2,000.

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EXAMPLE 2EXAMPLE 2

The daily water usage per person in Toledo, Ohio is normally distributed with a mean of 20 gallons and a standard deviation of 5 gallons.

About 68% of the daily water usage per person in Toledo lies between what two values?

± 1

That is, about 68% of the daily usage per person will lie between __________________ gallons.

Similarly for 95% and 99%, the intervals will be __________________________________________ .

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POINT ESTIMATESPoint estimate: one number (called a point) that is

used to estimate a population parameter.Examples of point estimates are the sample mean,

the sample standard deviation, the sample variance, the sample proportion, etc.

EXAMPLE: The number of defective items produced by a machine was recorded for five randomly selected hours during a 40-hour work week. The observed number of defectives were 12, 4, 7, 14, and 10. So the sample mean is ____ . Thus a point estimate for the weekly mean number of defectives is 9.4.

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INTERVAL ESTIMATESInterval Estimate: states the range within

which a population parameter probably lies.The interval within which a population

parameter is expected to occur is called a confidence interval.

The two confidence intervals that are used extensively are the 95% and the 99%.

A 95%confidence interval means that about 95% of the similarly constructed intervals will contain the parameter being estimated.

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INTERVAL ESTIMATES (continued)

Another interpretation of the 95% confidence interval is that 95% of the sample means for a specified sample size will lie within 1.96 standard deviations of the hypothesized population mean.

For the 99% confidence interval, 99% of the sample means for a specified sample size will lie within 2.58 standard deviations of the hypothesized population mean.

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Determining Sample Size for Probability SamplesFinancial, Statistical, and Managerial Issues

The larger the sample, the smaller the sampling error, but larger samples cost more.

Budget AvailableRules of Thumb

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Typical Sample Sizes

Number of

Consumer research

Business research*subgroup

National

Special

National

Specialanalyses

population

population

population

population  None/few

200-500

100-500

20-100

20-50 Average

500-1000

200-1000

50-200

50-100 Many

1000-2000

500-1000

200-500

100-250

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Sample Size DeterminationSample size depends on

Allowable Error/level of precision/ sampling error (E)Acceptable confidence in standard errors (Z)Population standard deviation ()

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Sample size determination

Problem involving means:Sample Size (n) = Z2 2 / E2

where:Z = level of confidence expressed in

standard errors = population standard deviationE = acceptable amount of sampling error

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Sample size determination

Problem involving proportions:Sample Size (n) = Z2 [P(1-P)] / E2

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Sampling Exercise

Let us assume we have a population of 5 people whose names and ages are given below:Abe 24Bob 30Cara 36Don 42Emily 36

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Average of all samples of size = 1

Abe 24 Bob 30Cara 36 Don 42 Emily 48

Average of all possible “size = 1”

samples= 36

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Average of all samples of size = 2

Abe, Bob (24+30)/2 = 27 Abe, Cara 30 Abe, Don 33 Bob, Cara 33 Abe, Emily 36 Bob, Don 36 Bob, Emily 39 Cara, Don 39 Cara, Emily 42Don, Emily 45Average of all possible “size = 2” samples=

36

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Average of all samples of size = 3

Abe, Bob, Cara 30

Abe, Bob, Don 32 Abe, Bob, Emily 34

Abe, Cara, Don 34 Abe, Cara, Emily 36 Bob, Cara, Don 36 Bob, Cara, Emily 38 Abe, Don, Emily 38 Bob, Don, Emily 40 Cara, Don, Emily 42Average of all possible “size = 3”

samples= 36

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Average of all samples of size = 4

Abe, Bob, Cara, Don 33Abe, Bob, Cara, Emily 34.5 Abe, Bob, Don, Emily 36Abe, Cara, Don, Emily 37.5 Bob, Cara, Don, Emily 39Average of all possible “size = 4”

samples= 36

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Average of all samples of size = 3

Abe, Bob, Cara, Don, Emily 36Average of all possible “size = 5”

samples= 36

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What can be learned?What is the average of the average of the

sample for a given size?Does the mean of any individual sample

equal to the population mean?Range of values for each sample size

category?

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Sampling DistributionPopulation distribution: A frequency

distribution of all the elements of a population.

Sample distribution: A frequency distribution of all the elements of an individual sample.

Sampling distribution- a frequency distribution of the means of many samples.

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Normal Distribution

Central Limit Theorem - Central Limit Theorem—distribution of a large number of sample means or sample proportions will approximate a normal distribution, regardless of the distribution of the population from which they were drawn

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The Standard Error of the Mean

Applies to the standard deviation of a distribution of sample means.

x =n√

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The Standard Error of the Distribution of ProportionsApplies to the standard deviation of a distribution of sample proportions.

Sampling Distribution of the Proportion

P (1-P)Sp =

n√

where:Sp = standard error of sampling distribution proportionP = estimate of population proportionn = sample size

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Sample size determination – adjusting for population sizeMake an adjustment in the sample size if

the sample size is more than 5 percent of the size of the total population. Called the Finite Population Correction (FPC).

x =n√

N - n√ N - 1