- A bag contains a large number of balls, all of the same size and weight. The balls are coloured Red, Blue or Yellow.
Jasmine asks each child in a group of 150 children to close their eyes, select a ball from the bag and show it to her. The child then replaces the ball and repeats the process a second time.
If both balls are the same colour the child receives a prize.
The results are given in the table below.
| \backslashbox{2nd colour}{1st colour} | Red | Blue | Yellow | Total |
| Red | 31 | 11 | 18 | 60 |
| Blue | 8 | 10 | 9 | 27 |
| Yellow | 21 | 9 | 33 | 63 |
| Total | 60 | 30 | 60 | 150 |
Jasmine carries out a test, at the \(5 \%\) level of significance, to see whether or not the colour of the 2nd ball is independent of the colour of the 1st ball.
- Calculate the expected frequencies for the cases where both balls are the same colour.
The test statistic Jasmine obtained was 12.712 to three decimal places.
- Use this value to complete the test, stating the critical value and conclusion clearly.
With reference to your calculations in part (a) and the nature of the experiment, (c) give a plausible reason why Jasmine may have obtained her conclusion in part (b).