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 |
- 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 \(\mathrm { B } ( 6,0.313 )\) 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 |
- Show how the expected frequency for \(x = 4\) is calculated.
- Test at the \(5 \%\) significance level whether the scientist's belief is correct.
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