Bacteria Colonies: Poisson Distribution A pathologist has...

Bacteria Colonies: Poisson Distribution 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 $2.80$ 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. (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 $$ P(r)=\frac{e^{-\lambda} \lambda^{r}}{r !} $$ where $\lambda=2.80$ is the average number of bacteria colonies per field. Compute $P(r)$ for $r=0,1,2,3,4$, and 5 or more. (b) Compute the expected number of colonies $E=100 P(r)$ for $r=0,1,2,3,4$, and 5 or more. (c) Compute the sample statistic $\chi^{2}=\Sigma \frac{(O-E)^{2}}{E}$ and the degrees of freedom. (d) Test the statement that the Poisson distribution fits the sample data. Use a 5\% level of significance.

Bacteria Colonies: Poisson Distribution 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 $2.80$ 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.
(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
$$
P(r)=\frac{e^{-\lambda} \lambda^{r}}{r !}
$$
where $\lambda=2.80$ is the average number of bacteria colonies per field. Compute $P(r)$ for $r=0,1,2,3,4$, and 5 or more.
(b) Compute the expected number of colonies $E=100 P(r)$ for $r=0,1,2,3,4$, and 5 or more.
(c) Compute the sample statistic $\chi^{2}=\Sigma \frac{(O-E)^{2}}{E}$ and the degrees of freedom.
(d) Test the statement that the Poisson distribution fits the sample data. Use a 5\% level of significance.

Key Concepts

Poisson Distribution

The Poisson distribution is a probability model used to describe the number of times an event occurs within a fixed interval of time or space, assuming that these events occur with a known constant mean rate and independently of the time since the last event. It is characterized by its parameter ? (lambda), which represents the average number of occurrences, and it provides the probability mass function P(r) = (e^(-?) ?^(r))/(r!), where r is the count of events.

Expected Frequency Calculation

In statistical modeling, the expected frequency is obtained by multiplying the probability of an event (derived from a theoretical distribution, such as the Poisson distribution in this case) by the total number of observations. This provides an estimate of how many occurrences should be observed in each category if the model is correct, and is essential for conducting goodness-of-fit tests.

Chi-Squared Goodness-of-Fit Test

The Chi-Squared (?²) goodness-of-fit test is a statistical hypothesis test used to determine how well an observed frequency distribution matches an expected frequency distribution predicted by a specific theoretical model. The test statistic is calculated using the formula ?² = ?((O - E)²/E), where O are the observed frequencies and E are the expected frequencies. This test assesses whether any discrepancies between observed and expected values are due to random chance or indicate a model misfit.

Degrees of Freedom

Degrees of freedom in the context of the Chi-Squared test represent the number of independent pieces of information that go into estimating a parameter. It is calculated as the number of categories minus the number of parameters estimated from the data minus one. Degrees of freedom are crucial for determining the appropriate critical value from the Chi-Squared distribution used to assess the significance of the test statistic.

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about the validity of a claim based on sample data. It involves formulating a null hypothesis (often a statement of no effect or no difference) and an alternative hypothesis, and then using a test statistic to determine the probability of obtaining the observed data if the null hypothesis were true. In the context of fitting the Poisson distribution to sample data, hypothesis testing helps decide whether the observed discrepancies can be attributed to chance or indicate that the Poisson model may not be a good fit.

Transcript

00:01 The following is the solution to number 18, which is the goodness of fit test for a poisson distribution.

00:06 This involves like bacteria in the mouth or something, and the lambda, the mean, is 2 .8 or 2 .80.

00:13 And we're asked to find the probability that zero occur, whatever it is.

00:18 I think it's like bacteria colonies or something, zero.

00:21 Probability that one occurs, two, three, four, and then greater than or equal to five.

00:26 So you can use the formula, but i like to use the software.

00:29 So i'm going to use the ti -84.

00:30 And if you go to second vars, which is the distributions, we can go to this posan pdf.

00:35 That stands for the probability density function.

00:37 And then we're just going to type in the mean, which is mu, in this case it lambda.

00:41 They call it lambda for the plusan.

00:43 And then the x value is just what i'm trying to find.

00:45 So this is the probability of zero, given that the mean is 2 .8.

00:50 And that gives us 0 .0608.

00:57 Okay, i'll do that one more time.

01:00 And then the other ones i'll just copy down.

01:02 So then the poisson pdf 2 .8 and then this time i'm going to make it 1.

01:09 And then we press enter and then it's 0 .1703 let's say.

01:13 So 0 .1703.

01:18 Okay, so that's what you're going to do and then you're going to change it to 2, 3, 4.

01:22 So i'll just go ahead and copy these down.

01:23 Whenever you do that, for the 2, you should get 0 .2384.

01:27 For the three you should get 0 .2225 and then for the probability of four you should get 0 .157.

01:37 Now for greater than equal to five, it's one minus the things that you just found.

01:42 So what you could do is you could just add 0, 1, 2, 3, and 4 together these probabilities, and you take 1 minus that and that'll give you the probability that it's greater and equal to 5 because we don't have enough time to do probability of 5 plus probability 6, 7, 8, 9, all the way up to infinity.

01:56 We don't have enough time.

01:58 So instead, it's quicker if we just take the total, which is one, minus the probability is that we don't want, which is zero through four.

02:05 Now, fortunately, there is a function on the calculator that adds it up for you.

02:10 If you want to use it, you don't have to.

02:13 But it's called the poisson cdf.

02:15 Cdf stands for the cumulative density function.

02:17 So it's one minus the poisson cdf.

02:20 So if we go second distribution, and you may have seen it already, it's the poisson cdf.

02:28 Actually, you know what? i'm going to second quit because i forgot to do one minus.

02:32 So one minus posseigne cdf.

02:36 And remember the mean was 2 .8.

02:39 Now instead of the x value being, i'm not going to say 0, 1, 2, 3, 4, i'm just going to say 4.

02:44 So the calculator automatically knows that it's 0 plus 1 plus 2 plus 3 plus 4.

02:48 The probabilities associated with that.

02:51 And we paste it in there, and that gives us 0 .152 .3.

02:55 Okay, so let's write that down.

02:59 So point equals 0 .1523.

03:04 So those are all the probabilities.

03:06 The second part of this, so part b, it says find the expected value if the sample size in was 100.

03:11 Now that's kind of nice because the math is super easy...

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