statistical methods introduction to estimation noha hussein elkhidir16/04/35

Statistical Methods

Introduction to Estimation

noha hussein elkhidir16/04/35

Estimation…

There are two types of inference: estimation and hypothesis testing; estimation is introduced first.

The objective of estimation is to determine the approximate value of a population parameter on the basis of a sample statistic.

E.g., the sample mean ( ) is employed to estimate the population mean ( ).


Some Concepts

*Estimators: Random variables used to estimate

population parameters.

Example:

p^ is an estimator of p

*Estimates: Specific values of an estimator.

Estimation…

The objective of estimation is to determine the approximate value of a population parameter on the basis of a sample statistic.

There are two types of estimators:

Point Estimator

Interval Estimator


Point Estimator…

A point estimator draws inferences about a population by estimating the value of an unknown parameter using a single value or point.

We saw earlier that point probabilities in continuous distributions were virtually zero. Likewise, we’d expect that the point estimator gets closer to the parameter value with an increased sample size, but point estimators don’t reflect the effects of larger sample sizes. Hence we will employ the interval estimator to estimate population parameters…


Interval Estimator…

An interval estimator draws inferences about a population by estimating the value of an unknown parameter using an interval.

That is we say (with some ___% certainty) that the population parameter of interest is between some lower and upper bounds.


Point & Interval Estimation…

For example, suppose we want to estimate the mean summer income of a class of business students. For n=25 students,

is calculated to be 400 $/week.

point estimate interval estimate

An alternative777 statement is:

The mean income is between 380 and 420 $/week.


Qualities of Estimators…Statisticians have already determined the “best” way to estimate a population

parameter. Qualities desirable in estimators include unbiasedness,

consistency, and relative efficiency:• An unbiased estimator of a population parameter is an

estimator whose expected value is equal to that parameter.• An unbiased estimator is said to be consistent if the

difference between the estimator and the parameter grows smaller as the sample size grows larger.

• If there are two unbiased estimators of a parameter, the one whose variance is smaller is said to be relatively efficient.


Confidence Interval Estimator for :

The probability 1– is called the confidence level.

lower confidence limit (LCL)

upper confidence limit (UCL)

Usually represented with a “plus/minus”

( ± ) sign

noha hussein elkhidir/ /١٤٤٤ ٠٩ ٣٠

Graphically…

…the actual location of the population mean …

…may be here… …or here……or possibly even

here…

The population mean is a fixed but unknown quantity. Its incorrect to interpret the confidence interval estimate as a probability statement about . The interval acts as the

lower and upper limits of the interval estimate of the population mean.noha hussein elkhidir16/04/35

Four commonly used confidence levels…Confidence Level


Example:

A computer company samples demand during lead time over 25 time periods:

Its is known that the standard deviation of demand over lead time is 75 computers. We want to estimate the mean demand over lead time with 95% confidence in order to set inventory levels…

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Example :

“We want to estimate the mean demand over lead time with 95% confidence in order to set inventory levels…”

Thus, the parameter to be estimated is the pop’n mean:

And so our confidence interval estimator will be:

IDENTIFY


Example :In order to use our confidence interval estimator, we need the following pieces of data:

therefore:

The lower and upper confidence limits are 340.76 and 399.56.

370.16

1.96

75

n 25 Given

Calculated from the data…


Interval Width…

A wide interval provides little information.

For example, suppose we estimate with 95% confidence that an accountant’s average starting salary is between $15,000 and $100,000.

Contrast this with: a 95% confidence interval estimate of starting salaries between $42,000 and $45,000.

The second estimate is much narrower, providing accounting students more precise information about starting salaries.


Interval Width…

The width of the confidence interval estimate is a function of the confidence level, the population standard deviation, and the sample size…


Interval Width…


A larger confidence level

produces a w i d e r

confidence interval:


Interval Width…


Larger values of

produce w i d e r

confidence intervals


Interval Width…


Increasing the sample size decreases the width of the confidence interval while the confidence level can remain unchanged.

Note: this also increases the cost of obtaining additional datanoha hussein elkhidir16/04/35

Selecting the Sample Size…

We can control the width of the interval by determining the sample size necessary to produce narrow intervals.

Suppose we want to estimate the mean demand “to within 5 units”; i.e. we want to the interval estimate to be:

Since:

It follows that

Solve for n to get requisite sample size!noha hussein elkhidir16/04/35

Selecting the Sample Size…

Solving the equation…

that is, to produce a 95% confidence interval estimate of the mean (±5 units), we need to sample 865 lead time periods (vs. the 25 data points we have currently).


Sample Size to Estimate a Mean…

The general formula for the sample size needed to estimate a population mean with an interval estimate of:

Requires a sample size of at least this large:


Example :

A lumber company must estimate the mean diameter of trees to determine whether or not there is sufficient lumber to harvest an area of forest. They need to estimate this to within 1 inch at a confidence level of 99%. The tree diameters are normally distributed with a standard deviation of 6 inches.

How many trees need to be sampled?


Example 10.2…Things we know:

Confidence level = 99%, therefore =.01

We want , hence W=1.We are given that = 6.

1


Example :

We compute…

That is, we will need to sample at least 239 trees to have a

99% confidence interval of 1


statistical methods introduction to estimation noha hussein elkhidir16/04/35

Documents

parameter value

unknown parameter

herethe population mean

estimator of p

sample mean

interval acts

approximate value

mean income