9231 P41 - Jun 2025 - Q3 - 8 marks
Eggs in a supermarket are sold in boxes of six. A supermarket manager wishes to model the number of broken eggs in the boxes sold in the store. A random sample of 2000 boxes is taken and the number of broken eggs recorded. The observed frequencies are shown in the table below.
Number of broken eggs | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|---|
Observed frequency | 1844 | 143 | 11 | 0 | 1 | 0 | 1 |
(a) Use the data to estimate the probability that an egg is broken. Give your answer correct to 4 significant figures.
It is decided to carry out a goodness of fit test at the \(0.5 \%\) significance level to determine whether a binomial distribution fits the data.
The observed frequencies and the expected frequencies are given in the following table.
Number of broken eggs | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|---|
Observed frequency | 1844 | 143 | 11 | 0 | 1 | 0 | 1 |
Expected frequency | 1831.3 | \(a\) | 6.016 | 0.119 | 0.001 | 0.000 | 0.000 |
(b) Show that \(a=162.6\) correct to 1 decimal place.
(c) Carry out a goodness of fit test at the \(0.5 \%\) level of significance to test whether a binomial distribution is a satisfactory model for the data.
(d) Give a reason why a binomial distribution may not be a suitable model in this situation.