 A Characteristic is a measurable description of an individual such as height, weight or a count meeting a certain requirement.  A Parameter is a numerical summary of a characteristic from an entire population such as mean, proportion or standard deviation.  A Statistic is a numerical summary of a characteristic from a sample taken from an entire population such as mean, proportion or standard deviation.

 A Point Estimate is the Statistic from experimental data that estimates the Parameter.  The confidence Interval is an interval that contains the Parameter (with some level of confidence) and is based on the Point Estimate.  Confidence Level is the expected proportion of intervals that will contain the Parameter.  Margin of Error is distance from the Point Estimate to the ends of the Confidence Interval (the possible error of the Point Estimate.

 To Find the Confidence Interval of a Proportion, the Z-distribution can be used to find the Critical Value IF the requirements for using the Normal Approximation of a Binomial are met (Section 8.2).  For a Proportion the experiment consists of a sample size (n) and the count (x) from the sample that meets the criteria producing the point estimate of p ( ) and a confidence Level and. If given, then.

 The Margin of Error is the (Standard Deviation) *(Critical Value).  Find the Critical Value using the Z- Distribution as in Chapter 7: InvNorm( ).  From 8.2, recall that the standard deviation of p is given by. So the Margin of Error is calculated by.  So the Confidence Interval is.  Given x, n, and a Confidence Level, find Confidence Intervals. Also do for n and.  Use 1-PropZInt.

 If a certain Margin of Error (E) is required, a sample size must be calculated.  If a is available, then the sample size needed to get a value of E or less is  If is not known, then the sample size needed to get a value of E or less is

 If the Confidence Interval is given, the Margin of Error and the Point Estimate can be found from the interval maximum and minimum values:  Margin of Error = Half the distance between the two values  Point Estimate = The middle of the interval.

 To find the confidence Interval for a Mean, find the Margin of Error (E) and add it to and subtract it from the point estimate :  The Margin of Error (E) is a factor (Critical Value) multiplied by the Standard Deviation:  or  The Critical Value depends on the Confidence Level desired.

 Find the Critical Value from the Confidence Level Required using the InvNorm( ).  Create the table of Confidence Levels and Critical Values for various Levels.  Then the Margin of Error is.  And the Confidence Interval is.  Given an, a and a required Confidence Level find the Critical Value, the margin of Error and the Confidence Interval.

 If the population standard deviation is unknown, the sample standard deviation must be used.  But for the sample standard deviation we can not use the Z-table to find the Critical Value. The sample standard deviation is probably not exact, so a wider different distribution is needed.  The larger the sample size the better estimate the sample standard deviation is of the population standard deviation.  Therefore the Critical Value will depend on the Confidence Level AND THE SAMPLE SIZE.

 The distribution used for the Mean with an unknown Standard Deviation is the Student or t Distribution and will be found in the t-table.  The columns of the table are the from the Confidence Level. The rows are the Degrees of Freedom and are one less than the sample size.  Find for several sample sizes and Confidence Levels.  Also use InvT(, n-1).

 Note that the t-distribution is symmetric like the Z distribution but wider.  Note that the t-distribution is not one distribution but many – one for each sample size. The smaller the sample size the wider the distribution.  As the sample size approaches infinity the t distribution approaches the Z distribution.

 After finding the Critical Value, the margin of Error and Confidence Interval can now be calculated as was done with the situation with a known standard deviation.  Then the Margin of Error is.  And the Confidence Interval is.  Given an, an s and a required Confidence Level find the Critical Value, the margin of Error and the Confidence Interval.  Also use Tinterval.

 Finding the Confidence Interval for a Standard Deviation (s) or Variance ( ) is different than the others have been.  There is no Error to be calculated.  The Critical Values comes from a different distribution that is not normal.  The (called Chi-squared) Distribution is skewed right, starts at 0 on the left and goes to on the right.  There are two Critical Values that are both positive.

 The columns are split into two halves. The half on the right are for - Chi-squared Right and use the value of. The half on the left are for - Chi-squared Left and use the value of.  The rows are again the Degrees of Freedom (n - 1).

 To find the Confidence Interval of a Variance use:  To find the Confidence Interval of a Standard Deviation use:  There are no calculator functions  Must be given s, n and a Confidence Level.