4 The score on one spin of a 5 -sided spinner is denoted by the random variable \(X\) with probability distribution as shown in the table.
| \(x\) | 0 | 1 | 2 | 3 | 4 |
| \(\mathrm { P } ( X = x )\) | 0.1 | 0.2 | 0.4 | 0.2 | 0.1 |
- Show that \(\operatorname { Var } ( X ) = 1.2\).
The spinner is spun 200 times. The score on each spin is noted and the mean, \(\bar { X }\), of the 200 scores is found. - Given that \(\mathrm { P } ( \bar { X } > a ) = 0.1\), find the value of \(a\).
- Explain whether it was necessary to use the Central Limit theorem in your answer to part (b).
- Johann has another, similar, spinner. He suspects that it is biased so that the mean score is less than 2 . He spins his spinner 200 times and finds that the mean of the 200 scores is 1.86 .
Given that the variance of the score on one spin of this spinner is also 1.2 , test Johann's suspicion at the 5\% significance level.