Chapter 10
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.
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
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…
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
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 alternative statement is:
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.
Confidence Interval Estimator for :
The probability 1– is called the confidence level.
lower confidence limit (LCL)
upper confidence limit (UCL) Usually represented
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
Four commonly used confidence levels…
Confidence Level
Example 10.1…
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…
Example 10.1…
“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:
Example 10.1…
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 GivenInterval 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.
Interval Width…
Interval Width…
The width of the confidence interval estimate is a function of the confidence level, the population standard deviation, and the sample size…
A larger confidence level produces a w i d e r
Interval Width…
The width of the confidence interval estimate is a function of the confidence level, the population standard deviation, and the sample size…
Larger values of produce w i d e r
Interval Width…
The width of the confidence interval estimate is a function of the confidence level, the population standard deviation, and the sample size…
Increasing the sample size decreases the width of the
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
Selecting the Sample Size…
Solving the equation…
Sample Size to Estimate a Mean…
The general formula for the sample size needed to estimate a population mean with an interval estimate of:
Example 10.2…
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.
Example 10.2…
Things we know:
Confidence level = 99%, therefore =.01
Example 10.2…
We compute…
