Each of 165 newly manufactured items is examined and the number of scratches per item is recorded (the items are supposed to be free of scratches), yielding the following data:Let X = the number of scratches on a randomly chosen item, and assume that X has a Poisson distribution with parameter ?.Number ofscratchesper item 0 1 2 3 4 5 6 7Observedfrequency 25 34 37 29 19 8 8 5(a) Find an unbiased estimator of ? and compute the estimate for the data. [Hint: E(X) = ? for X Poisson, soE(X) = ?](Round your answer to two decimal places.)(b) What is the standard deviation (standard error) of your estimator? Compute the estimated standard error. [Hint: ?X2 = ? for X Poisson.] (Round your answer to three decimal places.)

Given that each of 165 newly manufactured items is examined and the number of scratches per item is recorded (the items are supposed to be free of scratches), yielding the following data

x freq x*f x^2*f

0 25 0 0

1 34 34 34

2 37 74 148

3 29 87 261

4 19 76 304

5 8 40 200

6 8 48 288

7 5 35 245

165 394 1480

Mean 2.387878788 8.96969697

Variance 3.267731864

We get E(x) = 2.388 and hence unbiased estimator of mu is 2.388

b) Variance = 3.268