Central Limit Theorem

There are 32 students in a class.

Each student rolls a fair die repeatedly, stopping when their total number of sixes is 4 Each student records the total number of times they rolled the die.

Estimate the probability that the mean number of rolls for the class is less than 27.2

A random sample of 150 observations is taken from a geometric distribution with parameter 0.3

Estimate the probability that the mean of the sample is less than 3.45

A courier delivers parcels. The random variable X represents the number of parcels delivered successfully each day by the courier where X ~ B (400, 0.64)

A random sample X₁, X₂,... X₁₀₀ is taken. Estimate the probability that the mean number of parcels delivered each day by the courier is greater than 257

A company produces two colours of candles, blue and white. The standard deviation of the burning times of the blue candles is 2.6 minutes and the standard deviation of the burning times of the white candles is 2.4 minutes.

Nissim claims that the mean burning time of blue candles is more than 5 minutes greater than the mean burning time of white candles.

A random sample of 90 blue candles is found to have a mean burning time of 39.5 minutes. A random sample of 80 white candles is found to have a mean burning time of 33.7 minutes.

Stating your hypotheses clearly, use a suitable test to assess Nissim’s belief. Use a 1% level of significance.

Explain how the hypothesis test in part (a) would be carried out differently if the variances of the burning times of candles were unknown.

The burning times for the candles may not follow a normal distribution.

Describe the effect this would have on the calculations in the hypothesis test in part (a). Give a reason for your answer.

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, , is calculated.

Assuming the die is fair and using a suitable approximation,

find, to 3 significant figures, the value of k such that P( < k) = 0.05

Explain the relevance of the Central Limit Theorem in part (a).

Alicia considers the following hypotheses:

H₀: The die is fair H: The die is not fair

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.