Camera for Cataract. The following problem is modified from Rosner (2010).
A camera has been developed to detect the presence of cataract more accurately. Using this camera, the gray level of each point (or pixel) in the lens of a human eye can be characterized into 256 graduations, where a gray level of 1 represents black and a gray level of 256 represents white. To test the camera, photographs were taken of 12 randomly selected normal eyes and 9 randomly selected cataractous eyes (the two groups consist of different people). The median gray level of each eye was computed over the 10,000+ pixels in the lens. The data are given below
Under normality assumption we want to test if there is a significant difference in the gray levels between cataractous and normal eyes.
(a) Can you find any statistical evidence showing that the two gray levels are significantly different? Write down the hypotheses, conduct the test and report the p-value.
(b) Pretend that a new study is to be conducted for which the sample size is needed. From previous experiments you adopt = 16 for both groups. What sample size is needed to detect the difference of 10 in a 0:05-level testing against the two sided alternative with power 1 = 90%.