The following data are the number of persons who were waiting in line at a fast food counter at 100 randomly selected times during the week. At the 0.05 level, test whether the population of these values could be Poisson distributed.
x = Number Number of
of persons Observations
0 24
1 24
2 18
3 20
4 10
5 4
100
To test whether the population of these values could be Poisson distributed, we can use the chi-square goodness-of-fit test.
The null hypothesis for this test is that the observed data follows a Poisson distribution. The alternative hypothesis is that the observed data does not follow a Poisson distribution.
To perform the chi-square goodness-of-fit test, we need to calculate the expected frequencies for each category under the assumption that the population follows a Poisson distribution.
The expected frequency for each category can be calculated using the Poisson distribution formula:
E(x) = (e^(-λ) * λ^x) / x!
where λ is the mean of the Poisson distribution, which can be estimated by the sample mean.
Let's calculate the expected frequencies:
x Observed Frequency (O) Expected Frequency (E)
0 24 E(0)
1 24 E(1)
2 18 E(2)
3 20 E(3)
4 10 E(4)
5 4 E(5)
>5 100 E(>5)
To determine the expected frequency for x=0, we need to estimate the mean (λ). The mean can be estimated as the sample mean:
λ = (0 * 24 + 1 * 24 + 2 * 18 + 3 * 20 + 4 * 10 + 5 * 4 + >5 * 100) / 100
Let's compute the estimated mean:
λ = (0 + 24 + 36 + 60 + 40 + 20 + >500) / 100
For simplicity, let's assume >5 is for values greater than 5.
λ = (0 + 24 + 36 + 60 + 40 + 20 + 100) / 100
λ = 280 / 100
λ = 2.8
Now, we can calculate the expected frequencies for each category using the Poisson distribution formula:
E(0) = (e^(-2.8) * 2.8^0) / 0!
E(1) = (e^(-2.8) * 2.8^1) / 1!
E(2) = (e^(-2.8) * 2.8^2) / 2!
E(3) = (e^(-2.8) * 2.8^3) / 3!
E(4) = (e^(-2.8) * 2.8^4) / 4!
E(5) = (e^(-2.8) * 2.8^5) / 5!
E(>5) = 1 - (E(0) + E(1) + E(2) + E(3) + E(4) + E(5))
To perform the chi-square goodness-of-fit test, we need to calculate the chi-square statistic:
Chi-square = Σ(O - E)^2 / E
where Σ represents the summation of the values.
We then compare this statistic to the critical value from the chi-square distribution with (k-1) degrees of freedom, where k is the number of categories.
We have 7 categories, so the degrees of freedom for this test will be (7 - 1) = 6.
If the calculated chi-square statistic is greater than the critical value, we reject the null hypothesis and conclude that the observed data does not follow a Poisson distribution.
Now, you can follow these steps to calculate the chi-square statistic and compare it to the critical value to reach a conclusion.