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 ( p ) and a confidence Level and p x . If given p , then x p * n . n The Margin of Error is the (Standard Deviation) *(Critical Value). Z Find the Critical Value using the Z1 2 ). Distribution as in Chapter 7: InvNorm( From 8.2, recall that the standard deviation p*q of p is given by n . So the Margin of Error is calculated by Z * pn* q . p*q pZ * So the Confidence Interval is n . Given x, n, and a Confidence Level, find p Confidence Intervals. Also do for n and . Use 1-PropZInt. 2 2 2 If a certain Margin of Error (E) is required, a sample size must be calculated. If a p is available, then the sample size needed to get a value of E or less is Z 2 p * q E If p is not known, then the sample size needed to get a value of E or less is Z 2 0.25 E If the Confidence Interval is given, the Margin of Error and the Point Estimate can be found from the interval maximum and minimum values: Maximum Minimum Margin of Error = 2 between the two values Point Estimate = interval. Maximum Minimum 2 Half the distance The middle of the 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 x : x E, x E The Margin of Error (E) is a factor (Critical Value) multiplied by the Standard Deviation: E CV . .* or E CV . .* s The Critical Value depends on the Confidence Level desired. Find the Critical Value Z 2 from the Confidence Level Required using the InvNorm( 1 2 ). Create the table of Confidence Levels and Critical Values for various Levels. Z * Then the Margin of Error is 2 . And the Confidence Interval is x Z 2 * . Given an x , 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 2 ' s from the Confidence Level. The rows are the Degrees of Freedom and are one less than the sample size n 1. t Find 2 for several sample sizes and Confidence Levels. Also use InvT( 1 2 , 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 t * s . 2 And the Confidence Interval is x t * s . Given an x , an s and a required Confidence Level find the Critical Value, the margin of Error and the Confidence Interval. Also use Tinterval. 2 Finding the Confidence Interval for a Standard Deviation (s) or Variance ( s 2) 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. 2 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 R2 - Chi-squared Right and use the value of 2 . The half on the left are for L2 - Chi-squared Left and use the value of 1 2 . The rows are again the Degrees of Freedom (n - 1). To find the Confidence Interval of a Variance n 1 s 2 n 1 s 2 use: , R2 L2 To find the Confidence Interval of a Standard Deviation use: 2 2 n 1 s n 1 s , 2 2 R L There are no calculator functions Must be given s, n and a Confidence Level.