The number of times a machine broke down each week was...

The number of times a machine broke down each week was observed over a period of 100 weeks and recorded in the accompanying table. It was found that the average number of breakdowns per week over this period was $2.1$. Test the null hypothesis that the population distribution of breakdown is Poisson. \begin{tabular}{lcccccc} \hline Number of breakdowns & 0 & 1 & 2 & 3 & 4 & 5 or more \\ \hline Number of weeks & 10 & 24 & 32 & 23 & 6 & 5 \\ \hline \end{tabular}

The number of times a machine broke down each week was observed over a period of 100 weeks and recorded in the accompanying table. It was found that the average number of breakdowns per week over this period was $2.1$. Test the null hypothesis that the population distribution of breakdown is Poisson.
\begin{tabular}{lcccccc}
\hline Number of breakdowns & 0 & 1 & 2 & 3 & 4 & 5 or more \\
\hline Number of weeks & 10 & 24 & 32 & 23 & 6 & 5 \\
\hline
\end{tabular}

Key Concepts

Poisson Distribution

A Poisson distribution models the number of events occurring in a fixed interval of time or space when these events happen with a known constant mean rate and independently of time since the last event. It's defined by a single parameter (?), which represents the average rate of events, and is often used as a model for count data.

Hypothesis Testing

Hypothesis testing involves making an initial assumption (the null hypothesis) about a population parameter or distribution, and then using sample data to determine whether there is enough evidence to reject that hypothesis. In this context, the null hypothesis states that the observed counts of machine breakdowns follow a Poisson distribution.

Chi-Squared Goodness-of-Fit Test

The chi-squared goodness-of-fit test is a statistical test used to determine how well observed sample frequencies fit an expected distribution. It compares the observed frequencies with the expected frequencies computed under the assumption that the null hypothesis is true, and evaluates whether any differences are due to random chance or indicate a poor fit.

Expected Frequency Calculation

Expected frequency calculation involves computing the number of occurrences expected in each category based on the theoretical distribution, here the Poisson distribution. Using the Poisson probability mass function with the given average rate (?), the expected probability for each outcome is multiplied by the total number of observations to obtain the expected counts.

Data Grouping

Data grouping, which may include combining categories with low expected frequencies, is a necessary step in chi-squared tests to satisfy the test assumptions, particularly that each expected frequency is sufficiently high (commonly at least 5). This ensures the accuracy and validity of the chi-squared approximation.

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