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

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 1 if X is too small or too large The critical values are obtained from the Binomial distribution table. 12/6/2018 SA3202, Lecture 12

Some Binomial Tables Table 1 n=5, p=.1 === ====================================================== x 0 1 2 3 4 5 Probability 0.59049 0.32805 0.07290 0.00810 0.00045 0.00001 ========================================================= Table 2, n=5, p=.5 ========================================================== x 0 1 2 3 4 5 Probability 0.03125 0.15625 0.31250 0.31250 0.15625 0.03125 =========================================================== Table 3, n=10, p=.4 ============================================================= x 0 1 2 3 4 5 6 7 8 9 10 Prob. .0061 .0403 .1209 .2150 .2508 .2007 .1115 .0425 .0106 .0016 .0001 12/6/2018 SA3202, Lecture 12

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

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

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

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

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