9231 P23 - Jun 2018 - Q11O - 14 marks
Question 11 OR alternative.
A scientist carries out an experiment to investigate the quantity \(X\), which takes the values \(0,1,2,3,4,5\) or \(6\). He believes that the values taken by \(X\) follow a binomial distribution. He conducts \(250\) trials. His results are summarised in the following table.
| \(x\) | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|---|
| Observed frequency | 22 | 83 | 72 | 53 | 17 | 3 | 0 |
(i) Show that unbiased estimates of the mean and variance for these results are \(1.876\) and \(1.266\) respectively, correct to 3 decimal places. By evaluating the mean and variance of the distribution \(B(6,0.313)\), explain why \(X\) could have this distribution.
The expected frequencies corresponding to the distribution \(B(6,0.313)\) are shown in the following table.
| \(x\) | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|---|
| Observed frequency | 22 | 83 | 72 | 53 | 17 | 3 | 0 |
| Expected frequency | 26.3 | 71.9 | 81.8 | 49.7 | 17.0 | 3.1 | 0.2 |
(ii) Show how the expected frequency for \(x=4\) is calculated.
(iii) Test at the \(5\%\) significance level whether the scientist's belief is correct.