Begin2.DVI

It would also be nice to be able to adjust γ 1 and γ 2 so that one could say that there

is a 90% probability that the true mean lies within the specified limits. It would be

better still if one could change the 90% value to obtain limits for say a 95%,97%,or

99% probability that the true mean lie within the bounds specified. The probability

values 90% ,95% ,97% or 99% are called confidence levels associated with the calculated

mean value. To determine such limits one can employ the central limit theorem from

statistics which says that if (i) the number nof independent random variables in each

sample (the sample size) is large with a finite mean and variance for each sample

and (ii) the number of samples taken is large. Then the mean value associated with

the large set of sample means will be normally distributed.

Another way to state the above is as follows. For X a continuous random variable

which comes from some kind of probability distribution having a well defined mean

μand variance σ^2 , the central limit theorem states that if a large number of sample

means are collected , and one forms a table of these mean values and does an analysis

of the collected set of nmeans and forms a frequency table, just like table 11.3, then

one finds that these sample means are approximately normally distributed. The

central limit theorem also states that the distribution of the sample means can be

made as close to a normal distribution as desired, by taking larger and larger sample

sizes. It can be shown that the distribution of the sample means X ̄is approximately

normal with mean μand standard deviation σ/

√

n. The normal distribution can be

scaled to standard form by making an appropriate change of variables.

To use the central limit theorem select a confidence level γ= 1 −αwhich rep-

resents the area between the limits −zα/ 2 and zα/ 2 associated with the normalized

normal probability density function as illustrated in the figure 11-13. This deter-

mines values αand α/ 2 and the values ±zα/ 2 can be obtained from the normalized

probability table 11.5. Some example values for 1 −αare

1 −α .90 .95 .99 .999

α .10 .05 .01 .001

α/ 2 .05 .025 .005 .0005

zα/ 2 1.645 1.960 2.576 3.291

If xis the mean of a sample {x 1 , x 2 ,... , x n} of size n, then confidence limits on

the value xare determined as follows.