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Confidence Intervals

The Summary from class is outlined below.  It outlines our textbook, " Lind. (2005). Statistical techniques in business & economics (11th ed). New York: McGraw-Hill, Chapter 8."

A confidence interval is a range of values within which the population parameter is expected to occur.

Confidence Intervals define the upper and lower limits of a population mean calculated from a sample of the population. This "guess at the true mean" must be associated with a percentage of confidence to give the inference true meaning. The science of inferential statistics is based on this concept.

The factors that determine the width of a confidence interval are:

  1. The sample size, n.
  2. The variability in the population, usually estimated by s.
  3. The desired level of confidence.

If the population standard deviation is known or the sample is greater than 30 we use the z distribution.

Two Sided based on a sample.

If the population standard deviation is unknown and the sample is less than 30 we use the t distribution.

Confidence Intervals are calculated by the following equation:


Two sided based on a population.

The standard deviation divided by the square root of the number of samples is multiplied by the Z value.  The Z value is related to the Normal distribution curve and is the heart of inferential statistics. It is the X axis scalar that correlates to the probability of the entire distribution.

Here is a link found by MARGARET HOOPER (UofPhx, QNT 531, 2005); "Margin of Error and Confidence Levels Made Simple" http://www.isixsigma.com/library/content/c040607a.asp 

As the Confidence level increases the band of acceptable dispersion becomes wider. 

  • 80% Confidence level was +/- 0.257 

  • 95% confidence level was +/- 0.392 

  • 97% " " " +/- 0.420 

  • 99% " " " +/- 0.515 

  • 99.7% " " " +/-0.590 

This makes sense because you must allow for a greater variance of data to have higher confidence. This corresponds to the normal curve in the book.

By Michael King, UoPhx QNT 531, 2005

 


Point Estimates:

A point estimate is a single value (statistic) used to estimate a population value (parameter).


Interval Estimates:

An Interval 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 most extensively in our book are the 95% and the 99%.

  1. For a 95% confidence interval about 95% of the similarly constructed intervals will contain the parameter being estimated. Also 95% of the sample means for a specified sample size will lie within 1.96 standard deviations of the hypothesized population mean.
  2. 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.

Now we can always use these values (when asked) without looking them up in the table.


Standard Error of the Sample Mean

The standard error of the sample mean is the standard deviation of the sampling distribution of the sample means. It is computed by

is the symbol for the standard error of the sample mean.

is the standard deviation of the population.

n is the size of the sample.

If the standard deviation of the population is not known and n > or = to 30, the standard deviation of the sample, designated s, is used to approximate the population standard deviation. The formula for the standard error is:


Finite-Population Correction Factor

A population that has a fixed upper bound is said to be finite.

For a finite population, where the total number of objects is N and the size of the sample is n. 

An adjustment is made to the standard errors of the sample means and the proportion:

Standard error of the sample means

This adjustment is called the finite-population correction factor. If n/N < .05, the finite-population correction factor is ignored.

Standard error of the sample proportions


Selecting a Sample Size

To determine the size of a sample you should determine:

  1.  The degree of confidence selected.

  2. The maximum allowable error.

  3. The variation in the population.

Do not need to know the size of the population.

n = ( ((z x s)^2)/E)

E is the allowable error, z is the z- value corresponding to the selected level of confidence, and s is the sample deviation of the pilot survey.

 

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