Probability

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