(a)[2]
Show that the value of $\mathrm{Var}(X)$ is $1.2$.
(b)[4]
The spinner is turned $200$ times. The score for each turn is recorded and the mean, $\overline{X}$, of the $200$ scores is calculated. Given that $P(\overline{X} > a) = 0.1$, determine $a$.
(c)[1]
Explain whether the Central Limit Theorem needed to be used in your response to part (b).
(d)[5]
Johann has another spinner of a similar type. He thinks that it is biased so that the mean score is less than $2$. He spins his spinner $200$ times and obtains a mean of $1.86$ for the $200$ scores. 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.