1. Lecture 12 Binomial Tests and Quantiles
Outline of Today
Binomial Tests
Median
Quantiles
12/6/2018
SA3202, Lecture 12

2. Binomial Tests
Suppose X~Binom(n,p), p unknown. We are interested in testing a hypothesis about p. We shall consider the following cases:
1 H0: p=p0, vs H1 : p not equals p0
2 H0: p=p0, vs H1: p>p0
3 H0: p=p0, vs H1: p<p0
4 H0: p<=p0, vs H1: p>p0
5 H0: p>=p0, vs H1: p<p0
For small sample sizes, the test statistic is X itself:
For cases 2 and 4, Reject H0 if X is too large
3 and 5, if X is too small
if X is too small or too large
The critical values are obtained from the Binomial distribution table.
12/6/2018
SA3202, Lecture 12

3. Some Binomial Tables
Table n=5, p=.1
=== ====================================================== x
Probability
=========================================================
Table 2, n=5, p=.5
========================================================== x
Probability
===========================================================
Table 3, n=10, p=.4
============================================================= x
Prob
12/6/2018
SA3202, Lecture 12

4. For large sample sizes, we use the Normal approximation to the Binomial distribution
X~ AN(np0, np0(1-p0)) under H0
With a continuous correction:
For cases 2 and 4 , Reject H0 when X>=np0+.5+ table x s.e.
cases 3 and 5, Reject H0 when X<=np0-.5-table x s.e.
case 1, Reject H0 when X>=np0+.5+ table x s.e.
or when X<= np0-.5-table x s.e.
12/6/2018
SA3202, Lecture 12

5. Median
Definition Let X is a continuous r.v. with cdf F and pdf f. The median of (the distribution of ) X is defined as x0 by
Pr(X<= x0) =.5, or F(x0)=.5
Since X is continuous, Pr(X=x0)=0. Thus,
Pr(X<x0)=Pr(X>x0)=.5
12/6/2018
SA3202, Lecture 12

6. Remark 1: The median is a measure of the center of the distribution of X It is a more appropriate measure of the “center” of the distribution of X than the mean.
Remark 2: Many nonparametric methods focus on the median--- rather than the mean---as the basic characteristic of a distribution (population). Roughly speaking, in nonparametrics, the median plays a
Same role as the mean plays in classical methods. When the distribution is symmetric, the median and the mean coincide.
12/6/2018
SA3202, Lecture 12

7. Quantiles
Definition The quantile of order p, or the p-th quantile of (the distribution of ) X, denoted by xp , is defined as
Pr (X<= xp)=p, i.e, F(xp)=p
Some special cases:
1. The median is
12/6/2018
SA3202, Lecture 12

8. 2. The quartiles are
3. The deciles are
4. The percentiles are
12/6/2018
SA3202, Lecture 12