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Random variable Y has binomial distribution with n = 20. The sample median seperates the distribution 鈥� 50% are above 25, the probability in binomial distribution should be p = 0.5

The test id upper tailed so the value P-value is,

\(\begin{array}{l}P = P\left( {Y \ge 15} \right) = 1 - B(14;20,5)\\ = 0.021\end{array}\)

Cumulative density function cdf of bionomial random variable X with parameters n and p is

\(\begin{array}{l}B\left( {x;n,p} \right) = P\left( {X \le x} \right) = \sum\limits_{y = 0}^x {b\left( {y;n,p} \right),} \\x = 0,1...,n\end{array}\)

Theorem:

\(b\left( {x;n,p} \right) = \left\{ {\begin{array}{*{20}{c}}{\left( {\begin{array}{*{20}{c}}n\\x\end{array}} \right){p^x}{{\left( {1 - p} \right)}^{ - x}},x = 0,1,2...n}\\{0,otherwise}\end{array}} \right.\)

Hence the value is 0.021

There are total of 12 observation that exceeds 25, so, analogously as in (a), the P-value for the test can be computed as

\(\begin{array}{l}P = P\left( {Y \ge 12} \right) = 1 - B\left( {11;20,5} \right)\\ = 0.252\end{array}\)

At any reasonable significant level\(\alpha \)

P>\(\alpha \)

Do not reject the null hypothesis