Chapter 9.2, Problem 27PB

P-value for small samples Example 4, on whether dogs can detect bladder cancer by selecting the correct urine specimen (out of seven), used the normal sampling distribution to find the P-value. The normal distribution P-value approximates a P-value using the binomial distribution. That binomial P-value is more appropriate when either expected count is less than 15. In Example 4, n was 54, and 22 of the 54 selections were correct.

1. a. If H₀: p = 1/7 is true, X = number of correct selections has the binomial distribution with n = 54 and p = 1/7. Why?
2. b. For H_a: p > 1/7, with x = 22, the small sample P-value using the binomial is P(22) +P(23) +⋯+P(54), where P(x) denotes the binomial probability of outcome x with p = 1/7. (This equals 0.0000019.) Why would the P-value be this sum rather than just P(22)?