~Margin of Error~

~When a confidence interval is constructed for proportions or means, a standard normal curve is used (an exception is covered later in this discussion).

~The sample proportion,  (p cap) or the sample mean (x bar) is at the center of the standard normal curve (z score = 0).

~Then, the margin of error is the number of units from the center of this interval (p cap or x bar) to either end of the confidence interval.

~The units will be consistent with those displayed with the TI-83 when found.

~The levels of confidence will change the intervals for each different value (i.e., 90%, 95%, 98%, etc.).

~The larger the confidence level, the more sure we are that the interval contains our true proportion or mean. Consequently, the larger the confidence interval, the greater the distance from the center to each end. Consequently, the greater the margin of error.

~Z_a/2 is the positive Z value corresponding to the upper value of each confidence interval.

~This can be found by using 2nd Vars, menu 3, invnorm(1-a/2), where a = 1 - the level of confidence.  For example, the 90% confidence interval will yield an a of .10.

~Without the TI-83, the margin of error can be calculated by using the appropriate formula below:

~For proportions, E= Z_a/2 times √[(p cap)(q cap)] divided by √(n)

~For means, E= Z_a/2 times the population standard deviation s divided by the √(n).

~Using the TI-83, our work is much easier by using the following:

~For proportions, use STAT, Tests, down to menu A, enter, then enter x (number in our sample with the attribute in question), n (sample size), & confidence level (as a decimal). Be careful, since x must be entered as a whole number. If not a whole number, round up. For example, if 29% of 1025 adults use the internet for shopping, x would be (.29)(1025)=297.25 (which can not be used). So, we round up & use x=298.

~For means, use STAT, Tests, down to menu 7, enter, then enter the appropriate information.

~The TI-83 will display the confidence interval along with the sample proportion or the sample mean.

~This makes it easy to find the margin of error. Simple subtract the value of the sample proportion or sample mean from the upper value of the interval displayed.

~For estimating the population mean when the population standard deviation is unknown, use t-curves. Studies show you get better results this way. Some folks still resort to the standard normal curve, however.

~In that case, the TI-83 menu would be menu 8, under STAT, Tests, The sample standard deviation (S_x) is used in place of the unknown population standard deviation s.

~Note: Using the formulas, Z_a/2 is replaced by t_a/2. These critical values can be found in a table for the t distribution: Critical values (table A-3) or by the TI-83. To find the appropiate t-curve, use the degree of freedom (df) for that curve (n-1) along with the areas in the appropiate tail(s). These areas are usually given at the top or bottom of the table.

~Using the TI-83, go to Stats, Tests, menu 8 and enter 0 for the mean and √n for S_x along with the C-Level desired. The result will be a t confidence interval with the critical t-values at the end. This is based on the fact that a confidence interval using t-values may be written as (mean-E, mean+E), where E = t_a/2 times S_x/√n.

~For proportions, if the sample proportion (p cap) is unknown or not estimated, .5 is used. This places an upper bound on the margin of error & also assures us that our sample size is adequate when sampling.

~Also, in cases where the sample size n is ≥ .05 of the population size N, the finite population correction should be used.
(i.e., √[(N-n)/(N-1)] ).

~This factor is entered as an extra factor when entering the standard deviation (i.e., multiply the standard deviation entered by this factor).

~When this is done, the confidence intervals change along with the margin of error, for each case.