- A six-sided die has sides labelled \(1,2,3,4,5\) and 6
The random variable \(S\) represents the score when the die is rolled.
Alicia rolls the die 45 times and the mean score, \(\bar { S }\), is calculated.
Assuming the die is fair and using a suitable approximation,
- find, to 3 significant figures, the value of \(k\) such that \(\mathrm { P } ( \bar { S } < k ) = 0.05\)
- Explain the relevance of the Central Limit Theorem in part (a).
Alicia considers the following hypotheses:
\(\mathrm { H } _ { 0 }\) : The die is fair
\(\mathrm { H } _ { 1 }\) : The die is not fair
If \(\bar { S } < 3.1\) or \(\bar { S } > 3.9\), then \(\mathrm { H } _ { 0 }\) will be rejected.
Given that the true distribution of \(S\) has mean 4 and variance 3 - find the power of this test.
- Describe what would happen to the power of this test if Alicia were to increase the number of rolls of the die.
Give a reason for your answer.