With regard to John E. Freund\'s Mathematical Statistics with Applications, 7th

Question

With regard to John E. Freund's Mathematical Statistics with Applications, 7th edition. Chapter 12, Problem 4E. I'm having trouble understanding how the answer is derived from the binomial distribution table.

With reference to example 12.1...Suppose the manufacturer of a new medication want to test the null hypothesis theta = 0.90 against the alternative hypothesis theta = 0.60. His test statistic is X, in 20 trials, and he will accept the null hypothesis if x > 14: otherwise reject it. Find alpha and beta....from Table I ( Binomial Distribution)

Alpha = P(X<=14: theta = 0.90) = 0.0114

Beta = P(X>14: theta = 0.60) = 0.1255

Problem 4E refers to this example, but if you can explain how to use Table I (Bin. Dist. Table) to find the alpha and beta in this example, I should be able to work out 4E.

Explanation / Answer

The formula to calculate the binomial distribution is as shown below: -

n!/x!(n-x)!*p^x*q^n-x

In the above cases

Alpha = P(X<=14: theta = 0.90)

p=0.9, q=0.1, n=20, x=0 to 14. Hence the combined probability of this comes to 0.0114 (p is given at the top and n and x vertically in the table. To find the value, we need to combine those three values and look up the value that applies/corresponds to that)

Beta = P(X>14: theta = 0.60) = 0.1255

p=0.6, q=0.4, n=20, x=15 to 20. Hence the combined probability of this comes to 0.1255 (p is given at the top and n and x vertically in the table. To find the value, we need to combine those three values and look up the value that applies/corresponds to that)

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