Stats: Estimation
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Definitions
Confidence Interval
An interval estimate with a specific level of confidence
Confidence Level
The percent of the time the true mean will lie in the interval estimate given.
Consistent Estimator
An estimator which gets closer to the value of the parameter as the sample size increases.
Degrees of Freedom
The number of data values which are allowed to vary once a statistic has been determined.
Estimator
A sample statistic which is used to estimate a population parameter. It must be unbiased, consistent, and relatively efficient.
Interval Estimate
A range of values used to estimate a parameter.
Maximum Error of the Estimate
The maximum difference between the point estimate and the actual parameter. The Maximum Error of the Estimate is 0.5 the width of the confidence interval for means and proportions.
Point Estimate
A single value used to estimate a parameter.
Relatively Efficient Estimator
The estimator for a parameter with the smallest variance.
T distribution
A distribution used when the population variance is unknown.
Unbiased Estimator
An estimator whose expected value is the mean of the parameter being estimated.


Stats: Introduction to Estimation
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One area of concern in inferential statistics is the estimation of the population parameter from the sample statistic. It is important to realize the order here. The sample statistic is calculated from the sample data and the population parameter is inferred (or estimated) from this sample statistic. Let me say that again: Statistics are calculated, parameters are estimated.
We talked about problems of obtaining the value of the parameter earlier in the course when we talked about sampling techniques.
Another area of inferential statistics is sample size determination. That is, how large of a sample should be taken to make an accurate estimation. In these cases, the statistics can't be used since the sample hasn't been taken yet.
Point Estimates
There are two types of estimates we will find: Point Estimates and Interval Estimates. The point estimate is the single best value.
A good estimator must satisfy three conditions:
• Unbiased: The expected value of the estimator must be equal to the mean of the parameter
• Consistent: The value of the estimator approaches the value of the parameter as the sample size increases
• Relatively Efficient: The estimator has the smallest variance of all estimators which could be used
Confidence Intervals
The point estimate is going to be different from the population parameter because due to the sampling error, and there is no way to know who close it is to the actual parameter. For this reason, statisticians like to give an interval estimate which is a range of values used to estimate the parameter.
A confidence interval is an interval estimate with a specific level of confidence. A level of confidence is the probability that the interval estimate will contain the parameter. The level of confidence is 1 - alpha. 1-alpha area lies within the confidence interval.
Maximum Error of the Estimate
The maximum error of the estimate is denoted by E and is one-half the width of the confidence interval. The basic confidence interval for a symmetric distribution is set up to be the point estimate minus the maximum error of the estimate is less than the true population parameter which is less than the point estimate plus the maximum error of the estimate. This formula will work for means and proportions because they will use the Z or T distributions which are symmetric. Later, we will talk about variances, which don't use a symmetric distribution, and the formula will be different.
Area in Tails
Since the level of confidence is 1-alpha, the amount in the tails is alpha. There is a notation in statistics which means the score which has the specified area in the right tail.
Examples:
• Z(0.05) = 1.645 (the Z-score which has 0.05 to the right, and 0.4500 between 0 and it)
• Z(0.10) = 1.282 (the Z-score which has 0.10 to the right, and 0.4000 between 0 and it).
As a shorthand notation, the () are usually dropped, and the probability written as a subscript. The greek letter alpha is used represent the area in both tails for a confidence interval, and so alpha/2 will be the area in one tail.
Here are some common values
Confidence
Level Area between
0 and z-score Area in one
tail (alpha/2) z-score
50% 0.2500 0.2500 0.674
80% 0.4000 0.1000 1.282
90% 0.4500 0.0500 1.645
95% 0.4750 0.0250 1.960
98% 0.4900 0.0100 2.326
99% 0.4950 0.0050 2.576
Notice in the above table, that the area between 0 and the z-score is simply one-half of the confidence level. So, if there is a confidence level which isn't given above, all you need to do to find it is divide the confidence level by two, and then look up the area in the inside part of the Z-table and look up the z-score on the outside.
Also notice - if you look at the student's t distribution, the top row is a level of confidence, and the bottom row is the z-score. In fact, this is where I got the extra digit of accuracy from.

Stats: Estimating the Mean
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You are estimating the population mean, mu, not the sample mean, x bar.
Population Standard Deviation Known
If the population standard deviation, sigma is known, then the mean has a normal (Z) distribution.



The maximum error of the estimate is given by the formula for E shown. The Z here is the z-score obtained from the normal table, or the bottom of the t-table as explained in the introduction to estimation. The z-score is a factor of the level of confidence, so you may get in the habit of writing it next to the level of confidence.
Once you have computed E, I suggest you save it to the memory on your calculator. On the TI-82, a good choice would be the letter E. The reason for this is that the limits for the confidence interval are now found by subtracting and adding the maximum error of the estimate from/to the sample mean.

Student's t Distribution
When the population standard deviation is unknown, the mean has a Student's t distribution. The Student's t distribution was created by William T. Gosset, an Irish brewery worker. The brewery wouldn't allow him to publish his work under his name, so he used the pseudonym "Student".
The Student's t distribution is very similar to the standard normal distribution.
• It is symmetric about its mean
• It has a mean of zero
• It has a standard deviation and variance greater than 1.
• There are actually many t distributions, one for each degree of freedom
• As the sample size increases, the t distribution approaches the normal distribution.
• It is bell shaped.
• The t-scores can be negative or positive, but the probabilities are always positive.
Degrees of Freedom
A degree of freedom occurs for every data value which is allowed to vary once a statistic has been fixed. For a single mean, there are n-1 degrees of freedom. This value will change depending on the statistic being used.
Population Standard Deviation Unknown
If the population standard deviation, sigma is unknown, then the mean has a student's t (t) distribution and the sample standard deviation is used instead of the population standard deviation.
The maximum error of the estimate is given by the formula for E shown. The t here is the t-score obtained from the Student's t table. The t-score is a factor of the level of confidence and the sample size.
Once you have computed E, I suggest you save it to the memory on your calculator. On the TI-82, a good choice would be the letter E. The reason for this is that the limits for the confidence interval are now found by subtracting and adding the maximum error of the estimate from/to the sample mean.

Notice the formula is the same as for a population mean when the population standard deviation is known. The only thing that has changed is the formula for the maximum error of the estimate.