One of the earliest applications of the Poisson distribution was made by Student (1907) in studying errors made in counting yeast cells or blood corpuscles with a haemacytometer. In this study, yeast cells were killed and mixed with water and gelatin; the mixture was then spread on a glass and allowed to cool. Counts were made on 400 squares, and the data are summarized in the table after this problem. Here, the number of squares gives a count out of 400 of how squares contained each given number of cells. (a) Find an estimator for A as well as an asymptotic 95% confidence interval using the first moment of the Poiss () distribution. (b) Compare the observed number of squares to how many squares you would expect to find with each number of cells using the estimator from 1. (c) Use the second moment to propose another estimator for A that is different from the method of moments estimator in 1. (you may also need to use the first moment for this new estimator). Calculate the value of this new estimator from the data and the expected number of squares for each cell count. (d) Using your two estimators, plot the expected number of squares for each num- ber of cells, as well as true counts. Which estimator seems to fit the data better? (e) For each estimator, construct a hypothesis test for the hypotheses Ho A=0.7 vs H : \ 0.7 Do you reject the null hypothesis in each of them at level 0.05? # of cells # of squares expected values for 2. expected values for 3. 0 1 213 128 2 37 3 18 4 3 150 6 1 0