Class notes for ISE 201 San Jose State University Industrial & Systems Engineering Dept.

1. Probability & Statistics for Engineers & Scientists, byWalpole, Myers, Myers & Ye ~Chapter 9 Notes Class notes for ISE 201 San Jose State University Industrial & Systems Engineering Dept. Steve Kennedy

2. Unbiased Estimators • A statistic hat is an unbiased estimator of the parameter  if • Note that in calculating S2, the reason we divide by n-1 rather than n is so that S2 will be an unbiased estimator of 2. • Of all unbiased estimators of a parameter , the one with the smallest variance is called the most efficient estimator of . • Xbar is the most efficient estimator of . And,is called the standard error of the estimator Xbar

3. Confidence Intervals • When we use Xbar to estimate , we don't expect the estimate to be exact. A confidence interval is a statement that we are 100(1-)% confident that lies between two specified limits. • If xbar is the mean of a random sample of size n from a population with known variance 2, then is a 100(1-)% confidence interval for . • Here z/2 is the z value with area /2 to the right. • For example, for a 95% confidence interval,  = .05, and z.025 = 1.96. • If population not normal, this is still okay if n  30

4. Error of Estimate and Sample Size • If xbar is used as an estimate of , we can be 100(1-)% confident that the error of the estimate e will not exceed • It is possible to calculate the value of n necessary to achieve an error of size e. We can be 100(1-)% confident that the error will not exceed e when

5. One-Sided Confidence Bounds • Sometimes, instead of a confidence interval, we're only interested in a bound in a single direction. • In this case, a (1-)100% confidence bound uses z in the appropriate direction rather z/2 in either direction. • So the (1-)100% confidence bound would be eitherdepending upon the direction of interest.

6. Confidence Interval if  is Unknown • If  is unknown, the calculations are the same, using t/2 with  = n-1 degrees of freedom, instead of z/2, and using s calculated from the sample rather than . • As before, use of the t-distribution requires that the original population be normally distributed. • The standard error of the estimate (i.e., the standard deviation of the estimator) in this case is • Note that if  is unknown, but n  30, s is still used instead of , but the normal distribution is used instead of the t-distribution. • This is called a large sample confidence interval.

7. Difference Between Two Means • If xbar1 and xbar2 are the means of independent random samples of size n1 and n2, drawn from two populations with variances 12 and 22, then, if z/2 is the z-value with area /2 to the right of it, a 100(1-)% confidence interval for 1 - 2 is given by • Requires a reasonable sample size or a normal-like population for the central limit theorem to apply. • It is important that the two samples be randomly selected (and independent of each other). • Can be used if  unknown as long as sample sizes are large.

8. Estimating a Proportion • An estimator of p in a binomial experiment is Phat = X / n , where X is a binomial random variable indicating the number of successes in n trials. The sample proportion, phat = x / n is a point estimator of p. • What is the mean and variance of a binomial random variable X? • To find a confidence interval for p, first find the mean and variance of Phat:

9. Confidence Interval for a Proportion • If phat is the proportion of successes in a random sample of size n, and qhat = 1 - phat , then a (1-)100% confidence interval for the binomial parameter p is given by • Note that n must be reasonably large and p not too close to 0 or 1. • Rule of thumb: both np and nq must be  5. • This also works if the binomial is used to approximate the hypergeometric distribution (when n is small relative to N).

10. Error of Estimate for a Proportion • If phat is used to estimate p, we can be (1-)100% confident that the error of estimate will not exceed • Then, to achieve an error of e, the sample size must be at least • If phat is unknown, we can be at least 100(1-)% confident using an upper limit on the sample size of

11. The Difference of Two Proportions • If p1hat and p2hat are the proportion of successes in random samples of size n1 and n2, an approximate (1-)100% confidence interval for the difference of two binomial parameters is