A pathologist has been studying the frequency of bacterial colonies within the field of a microscope using samples of throat cultures from healthy adults. Long-term history indicates that there is an average of 2.82 bacteria colonies per field. Let r be a random variable that represents the number of bacteria colonies per field. Let O represent the number of observed bacteria colonies per field for throat cultures from healthy adults. A random sample of 100 healthy adults gave the following information. r 0 1 2 3 4 5 or more O 10 12 30 21 21 6 (a) The pathologist wants to use a Poisson distribution to represent the probability of r, the number of bacteria colonies per field. The Poisson distribution is given below. P(r) = e−λλr r! Here λ = 2.82 is the average number of bacteria colonies per field. Compute P(r) for r = 0, 1, 2, 3, 4, and 5 or more. (Round your answers to three decimal places.) P(0) = P(1) = P(2) = P(3) = P(4) = P(5 or more) =
(b) Compute the expected number of colonies E = 100P(r) for r = 0, 1, 2, 3, 4, and 5 or more. (Round your answers to one decimal place.) E(0) = E(1) = E(2) = E(3) = E(4) = E(5 or more) =
(c) Compute the sample statistic χ2 = (O − E)2 E and the degrees of freedom. (Round your test statistic to three decimal places.) d. f. = χ2 =
(d) Test the statement that the Poisson distribution fits the sample data. Use a 5% level of significance. Reject the null hypothesis. There is sufficient evidence to conclude the Poisson distribution does not fit. Reject the null hypothesis. There is insufficient evidence to conclude the Poisson distribution does not fit. Fail to reject the null hypothesis. There is sufficient evidence to conclude the Poisson distribution does not fit. Fail to reject the null hypothesis. There is insufficient evidence to conclude the Poisson distribution does not fit.