# Observed and expected frequencies

Please see the attached file for full problem description.

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? A sample of n independent observations are obtained of a random variable having a Poisson distribution with mean . Show that the maximum likelihood estimate of is he sample mean show that the corresponding estimator is an unbiased estimator of , and has variance .

The scientists Rutherford and Geiger reported an experiment in which they counted the number of alpha particles emitted from a radioactive source during intervals of 7.5 seconds duration, for 2612 different intervals. A total of 10126 particles were counted. The data obtained are summarised in the below.

Number 0 1 2 3 4 5 6 7 8 9 10 11 12 >12 Total

Frequency 57 203 383 525 532 408 273 139 49 27 10 4 2 0 2612

It has been suggested that the number of particles emitted in a n interval may be adequately modelled by a Poisson distribution. Assuming this conjecture to be correct, find the maximum likelihood estimate of the mean of this distribution, and use this to estimate the expected frequencies corresponding to the observed frequencies given in the table. Comment informally on the extent of agreement between these observed and expected frequencies.

© BrainMass Inc. brainmass.com October 24, 2018, 5:57 pm ad1c9bdddfhttps://brainmass.com/math/probability/observed-and-expected-frequencies-26221

#### Solution Summary

This shows how to find the maximum likelihood estimate of the mean of a Poisson distribution, and estimate the expected frequencies corresponding to observed frequencies.

Expected Frequencies and Goodness of Fit Test

A salesperson makes four calls per day. A sample of 100 days gives the following frequencies of sales volumes.

Number of Sales

0

1

2

3

4

Observed Frequency

30

32

25

10

3

Total 100

Records show sales are made to 30% of all sales calls. Assuming independent sales calls, the number of sales per day should follow a binomial distribution.

For this exercise, assume that the population has a binomial distribution with

n = 4, p = .30, and x = 0, 1, 2, 3, and 4.

a. Compute the expected frequencies for x = 0, 1, 2, 3, and 4 by using the binomial probability function. Combine categories if necessary to satisfy the requirement that the expected frequency is five or more for all categories.

b. Use the goodness of fit test to determine whether the assumption of a binomial distribution should be rejected. Use alpha = .05. Because no parameters of the binomial distribution were estimated from the sample data, the degrees of freedom are k - 1 when k is the number of categories.

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