91Ó°ÊÓ

A group of civil engineering students has tabulated the number of cars passing eastbound through the intersection of Mill and University Avenues. They obtained the data in the following table. (a) Does the assumption of a Poisson distribution seem appropriate as a probability model for this process? Use \(\alpha=0.05 .\) (b) Calculate the \(P\) -value for this test. $$\begin{array}{cccc}\begin{array}{c}\text { Vehicles per } \\\\\text { Minute }\end{array} & \begin{array}{c}\text { Observed } \\\\\text { Frequency }\end{array} & \begin{array}{c}\text { Vehicles per } \\\\\text { Minute }\end{array} & \begin{array}{c}\text { Observed } \\\\\text { Frequency }\end{array} \\\\\hline 40 & 14 & 53 & 102 \\\41 & 24 & 54 & 96 \\\42 & 57 & 55 & 90 \\\43 & 111 & 56 & 81 \\\44 & 194 & 57 & 73 \\\45 & 256 & 58 & 64 \\\46 & 296 & 59 & 61 \\\47 & 378 & 60 & 59 \\\48 & 250 & 61 & 50 \\\49 & 185 & 62 & 42 \\\50 & 171 & 63 & 29 \\\51 & 150 & 64 & 18 \\\52 & 110 & 65 & 15 \\\\\hline\end{array}$$

Step by step solution

01

Check Poisson Distribution Suitability

The Poisson distribution is often used to model the number of events occurring in a fixed interval. For the distribution to be appropriate, the mean and variance of the observed data should be approximately equal. Calculate the sample mean \[ \text{Mean (}\lambda\text{)} = \frac{\sum (x \times f)}{\sum f}\] and the sample variance \[ \text{Variance} = \frac{\sum ((x - \lambda)^2 \times f)}{\sum f - 1}\].Begin by calculating both values using the provided data.

02

Goodness-of-Fit Test Calculation

Use a Chi-square goodness-of-fit test to verify if there's a significant difference between observed frequencies and expected frequencies, assuming a Poisson distribution. The expected frequency for each category can be calculated using:\[ E(x) = N \times \frac{\lambda^x e^{-\lambda}}{x!} \]where \(N\) is the total number of observed data points and \(x!\) is the factorial of \(x\). Calculate these values for each category and compare them to the observed frequencies.

03

Critical Value and P-Value Determination

For a Chi-square test with \(df = k - 1 - p\) (k = number of categories, p = parameters estimated from the data, usually 1 for \(\lambda\)), determine the critical value for \(\alpha = 0.05\) using a Chi-square distribution table. Then calculate the test statistic \[ \chi^2 = \sum \frac{(O_i - E_i)^2}{E_i} \] where \(O_i\) is the observed frequency and \(E_i\) the expected frequency. Compare this statistic with the critical value to determine whether the Poisson distribution is an appropriate model.

04

Interpret P-Value

The P-value is the probability of observing a test statistic as extreme as, or more extreme than, the observed value under the null hypothesis. If the P-value is less than or equal to \(\alpha = 0.05\), reject the null hypothesis."

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Goodness-of-Fit Test

The Goodness-of-Fit Test is a statistical procedure used to determine how well a sample data matches a distribution from a population with a specific distribution. In this context, we're checking if the number of cars passing through an intersection fits a Poisson distribution.

This test involves comparing observed data with expected data under the assumed distribution. The Poisson distribution is applicable for rare events in a fixed interval, characterized by its parameter \( \lambda \), which equals the average rate of occurrence.

Here's how the process generally works:

• First, calculate the mean \( (\lambda) \) of the observed data. It's done by multiplying each event (e.g., vehicles per minute) by its recorded frequency and dividing by the total number of observations.
• Next, determine the variance of the observed data. For a Goodness-of-Fit Test to be reasonable, the mean and variance of a Poisson distribution should be approximately equal.
• Finally, compute the expected frequencies using the Poisson formula and compare these to the observed frequencies.

The extent to which observed frequencies deviate from expected frequencies indicates whether or not the Poisson distribution is a suitable model. If the differences seem large, the data might not fit well to the Poisson model.

Chi-Square Test

The Chi-Square Test is a method used in the context of the Goodness-of-Fit Test to determine whether there is a significant difference between observed and expected frequencies in one or more categories. This tool is crucial when examining the goodness of fit for complex models like the Poisson distribution.

To perform the Chi-Square Test, follow these steps:

• Calculate the expected frequency \( E(x) \) for each category using the Poisson formula \( E(x) = N \times \frac{\lambda^x e^{-\lambda}}{x!} \). Here, \( N \) is the total number of observations.
• Compute the Chi-Square statistic \( \chi^2 \) using the formula \( \chi^2 = \sum \frac{(O_i - E_i)^2}{E_i} \), where \( O_i \) and \( E_i \) are the observed and expected frequencies, respectively.
• Compare the calculated \( \chi^2 \) value with the critical value from the Chi-square distribution table based on degrees of freedom (\( df = k - 1 - p \), where \( k \) is the number of categories and \( p \) is the estimated parameters).

If the \( \chi^2 \) value is less than the critical value, then the observed data fits the expected model well. A relatively higher \( \chi^2 \) value leads to the conclusion that there is a significant discrepancy between observed and model-predicted frequencies, indicating a poor fit.

P-Value Calculation

P-Value Calculation is an essential part of the hypothesis testing in the context of the Chi-Square and Goodness-of-Fit Tests. It provides a probability measure that helps to determine whether observed data significantly deviate from the expected data under the null hypothesis.

The P-value represents the probability that the observed test statistic (or something more extreme) occurs if the null hypothesis is true. Here’s how it’s calculated and interpreted:

• After finding the \( \chi^2 \) value, the P-value is derived from the Chi-square distribution table based on the calculated degrees of freedom and the test statistic.
• The smaller the P-value, the stronger the evidence against the null hypothesis, implying that the model does not fit well.
• In our example, if the P-value is less than or equal to the significance level (e.g., \( \alpha = 0.05 \)), we reject the null hypothesis.

A high P-value suggests that the observed frequencies align well with the expected frequencies under the assumed distribution, affirming the suitability of the Poisson model in this context. Conversely, a low P-value indicates a lack of fit, prompting a reevaluation of the distribution's appropriateness for the data set. This systematic approach aids in making informed decisions about the statistical models applied to real-world scenarios.