Discrete uniform distribution sample mean

5 The score on one throw of a 4 -sided die is denoted by the random variable \(X\) with probability distribution as shown in the table.

1. Show that \(\operatorname { Var } ( X ) = 1.25\). The die is thrown 300 times. The score on each throw is noted and the mean, \(\bar { X }\), of the 300 scores is found.
2. Use a normal distribution to find \(\mathrm { P } ( \bar { X } < 1.4 )\).
3. Justify the use of the normal distribution in part (ii).

1. 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,

1. find, to 3 significant figures, the value of \(k\) such that \(\mathrm { P } ( \bar { S } < k ) = 0.05\)
2. 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
3. find the power of this test.
4. 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.

3 A discrete random variable \(X\) has the distribution \(\mathrm { U } ( 11 )\).
The mean of 50 observations of \(X\) is denoted by \(\bar { X }\).
Use an approximate method, which should be justified, to find \(\mathrm { P } ( \bar { X } \leqslant 6.10 )\).

A fair six-sided die is labelled with the numbers 1, 2, 3, 4, 5 and 6. The die is rolled 40 times and the score, \(S\), for each roll is recorded.

1. Find the mean and the variance of \(S\). [2]
2. Find an approximation for the probability that the mean of the 40 scores is less than 3 [3]