© aSup -2006 Probability and Normal Distribution 1 PROBABILITY.
The Normal Probability Distribution
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Transcript of The Normal Probability Distribution
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)MedianMode
Scores in our class
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 -
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.
Between:
68.26%
95.44%
99.97%
Between:
68.26%
95.44%
99.97%
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
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
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
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
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?
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 =.
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.
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 __________________________________________ .
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.
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.
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.
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
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
Sample Size DeterminationSample size depends on
Allowable Error/level of precision/ sampling error (E)Acceptable confidence in standard errors (Z)Population standard deviation ()
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
Sample size determination
Problem involving proportions:Sample Size (n) = Z2 [P(1-P)] / E2
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
Average of all samples of size = 1
Abe 24 Bob 30Cara 36 Don 42 Emily 48
Average of all possible “size = 1”
samples= 36
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
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
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
Average of all samples of size = 3
Abe, Bob, Cara, Don, Emily 36Average of all possible “size = 5”
samples= 36
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?
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.
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
The Standard Error of the Mean
Applies to the standard deviation of a distribution of sample means.
x =n√
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
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