1. Binomial distribution (Analysis of proportion)
Chapter 11
Binomial distribution
(Analysis of proportion)

2. Example: Apple scab disease
Twenty apples of the variety “Summer red” were collected at random from an old apple tree. Seven of the twenty apples had signs of the black or grey-brown lesions associated with apple scab. We wish to test the hypothesis that the proportion of apples with scab is 25%.

3. Binomial distribution
The number of apples with apple scab among 20 sampled apples is binomial, with size n = 20 and “success” probability p of finding apple cab, and observe Y = 7. If H0 is true then the probability of observing 7 apples with scab is P(Y = 7) = Thus, we do not reject the hypothesis that the proportion of apples with scab is 25%.

4. Confidence interval for proportion
The obvious estimate for that is obtained by counting the number Y = y of successes and then dividing by the size n. The normal approximation can be used for y/n, provided the number of observations, n, is large. The 95% confidence interval for p then becomes

5. Example: Apple scab disease
Our estimate of the proportion of apples infected with apple scab from this tree is The 95% confidence interval for the proportion of apples that are infected with scab is Hence we are 95% confident that the interval 14.10% to 55.90% contains the proportion of apples from the tree that are infected with apple scab disease.