9.15 Dogs and cancer A recent study4 considered whether dogs could be trained to detect whether a person has lung cancer or breast cancer by smelling the subject’s breath.

The researchers trained five ordinary household dogs to distinguish, by scent alone, exhaled breath samples of 55 lung and 31 breast cancer patients from those of 83 healthy controls. A dog gave a correct indication of a cancer sample by sitting in front of that sample when it was randomly placed among four control samples. Once trained, the dogs’ ability to distinguish cancer patients from controls was tested using breath samples from subjects not previously encountered by the dogs. (The researchers blinded both dog handlers and experimental observers to the identity of breath samples.) Let p denote the probability a dog correctly detects a cancer sample placed among five samples when the other four are controls.

a. Set up the null hypothesis that the dog’s predictions correspond to random guessing.

b. Set up the alternative hypothesis to test whether the probability of a correct selection differs from random guessing.

c. Set up the alternative hypothesis to test whether the probability of a correct selection is greater than with random guessing.

d. In one test with 83 Stage I lung cancer samples, the dogs correctly identified the cancer sample 81 times.

The test statistic for the alternative hypothesis in part c was z = 17.7. Report the P-value to three decimal places and interpret. (The success of dogs in this study made researchers wonder whether dogs can detect cancer at an earlier stage than conventional methods such as MRI scans.)