stats notes.pptx

7/28/2019 STATS NOTES.pptx

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Sampling Distributions and Confidence Intervals for the Mean

POINT ESTIMATION OF A POPULATION OR PROCESS PARAMETER

Defn: A parameter is a summary value for a population or process.

Defn: A sample statistic is a summary value for a sample.

To use a sample statistic as an estimator of a parameter, the sample must be a

random sample from a population or a rational group from the a process.

Eg

The mean and standard deviation of a population of measurements arepopulation parameters.

The mean and standard deviation of a sample of measurements aresample statistics


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Defn: A point estimator is the numeric value of a sample statistic that is used to

estimate the value of a population or process parameter.

Note: An estimator should be unbiased.

Defn: An unbiased estimator is a sample statistic whose expected value is equal to

the parameter being estimated.

Elimination of systematic bias is assured when the sample statistic is for a random

sample.


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Population parameter Estimator

Mean,

Variance,

Standard deviation,

Table for frequently used point estimators of population parameters.


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THE CONCEPT OF A SAMPLING DISTRIBUTION

Dfn: A population distribution is the distribution of all the individual measurements in

a population.

Dfn: Sample distribution is the distribution of the individual values included in a

sample

A sample statistic varies in value from sample to sample because of random sampling

variability, or sampling error.

Thus why any sample statistic is regarded as a variable whose distribution of values is

represented by a sampling distribution.


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SAMPLING DISTRIBUTION OF THE MEAN

The sampling distribution of the mean is described by determining the mean of

such a distribution, which is the expected value (), and the standard deviation ofthe distribution of sample means ().

is usually called the error of the mean.

When the population or process parameters are known, the expected value and

standard error for the sampling distribution of the mean are:

() = and =

.


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Example

Suppose the mean of a very large population is = 50.0 and the standarddeviation of the measurements is = 12. Determine the sampling distribution ofthe sample means for a sample size of = 36, in terms of the expected value andthe standard error of the distribution.


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Solution

= = 50.0

=

= 2.0


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THE CENTRAL LIMIT THEOREM

Theorem: As the sample size is increased, the sampling distribution of the

mean (and for other samples as well) approaches the normal distribution

in form, regardless of the form of the population distribution from which

the sample was taken.

ie, If

is sufficiently large ( 30

), then

~(,

)approximately.


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Example:

At the university, the mean age of the students is 22.3 years, and the standarddeviation is 4 years. A random sample of 64 students is drawn. What is the

probability that the average age of the students is greater than 23 years?


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Example:

An auditor takes a random sample of size = 36 from a population of 1000 accountsreceivable . The mean value of the accounts receivable for the population is =$260.00, with the population standard deviation = 45.00.

a) What is the probability that the sample mean will be less than$250.00?

b) What is the probability that the sample mean will be within $15.00 ofthe population mean?


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CONFIDENCE INTERVALS FOR THE MEAN USING THE NORMAL DISTRIBUTION

Dfn: A confidence interval for the mean is an estimate interval constructed with

respect to the sample mean by which the likelihood that the interval includes thevalue of the population mean can be specified.

The level of confidence associated with a confidence interval indicates the long-run

percentage of the such intervals which would include the parameter being

estimated.

When use of the normal probability distribution is warranted, the confidence

interval for the mean is determined by

or when the population is not known, by

.

Note: is the number of standard deviation units from the mean.


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In particular, (for unknown )

is a 68% confidence interval for .

1.645


1.96 is a 95% confidence interval for .

12.58



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Example:

For a given week, a random sample of 30 hourly employees selected from a very large

number of employees in a manufacturing firm has a sample mean wage of

= $180.00, with a sample standard deviation of = $14.00.Construct a 95% confidence interval for the mean wage for all hourly employees in the

firm.


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Solution: unknown ,

Therefore, 95% conf int. is given by 1.96

.

= 14, = 30

1.96

= 180 1.96

14

30

= ($174.98, $185.02)

Therefore, we can state that the mean wage level for all employees is between $174.98

and $185.02, with a 95% confidence in this estimate.


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THE DISTRIBUTION AND CONFIDENCE INTERVALS FOR THE MEAN

The distribution is a family of distributions, with a somewhat different distribution

associated with the degrees of freedom(df).

For a confidence interval for the population mean based on a sample of size n,

df = n 1.

For sample size n < 30.


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The degrees of freedom indicate the number of values that are in fact free to vary in

the sample that serves as the basis for the confidence interval

Where, df = n 1, the confidence interval for estimating the population mean when useof the t distribution is appropriate is

X t

sx

.


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Example:

The mean operating life for a random sample ofn = 10 light bulbs is X = 4000,with the sample standard deviation s = 200hr. The operating life of bulbs in generalis assumed to be approximately normally distributed. Estimate the mean operating

life for the population of bulbs from this sample using a 95 percent confidence

interval.


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Solution

Since n=10


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HYPOTHESIS TESTING CONCERNING THE VALUE OF THE

POPULATION MEAN

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