S3

Please can someone explain all of Q 7 on this paper?
Paper:http://pmt.physicsandmathstutor.com/download/Maths/A-level/S3/Papers-Edexcel/June%202015%20(IAL)%20QP%20-%20S3%20Edexcel.pdf

Markscheme: http://pmt.physicsandmathstutor.com/download/Maths/A-level/S3/Papers-Edexcel/June%202015%20(IAL)%20MS%20-%20S3%20Edexcel.pdf

Thank you

Part (a), you have a table of probabilities

Score 1 2 3 4 5 6
Prob 1/6 1/6 1/6 1/6 1/6 1/6

Work out the mean and variance from this table3.

Part (b), apply the central limit theorem to $\bar{S}$: you know that it is approximately normally distributed with mean = ? and variance = ? You can work out both ?'s from part (a).

Thank you for your reply, I did as you described for a, got mean correct but a for the variance using unbiased estimate formula, but i got 3.5 again? I don understand why the unbiased estimate formula doesn't work maybe I'm doing it wrong, i used n=6?

For b) I think I understand now, so because we dont know that its normally distributed thats why we have to put the variance over 40?

Perhaps the question is a little misleading (in introducing the sample of 40 straight away), but you're not estimating anything from a sample here! You don't want the unbiased estimator formula, you are calculating the variance of S, from its known probability distribution. So you want the formula

$\displaystyle \frac{1}{6} \sum_{i = 1}^{6} (i - 3.5)^2$



No, the central limit theorem tells you that $\bar{S}$ is approximately normally distributed with the same mean as S and with variance Var(S)/40.

Perhaps the question is a little misleading (in introducing the sample of 40 straight away), but you're not estimating anything from a sample here! You don't want the unbiased estimator formula, you are calculating the variance of S, from its known probability distribution. So you want the formula

$\displaystyle \frac{1}{6} \sum_{i = 1}^{6} (i - 3.5)^2$



No, the central limit theorem tells you that $\bar{S}$ is approximately normally distributed with the same mean as S and with variance Var(S)/40.

Could you also help me with Q 8, why is it binomial?

A single confidence interval either does, or does not, contain mu, with probability 0.9. So whether or not a single confidence interval contains mu is a Bernoulli random variable with probability of success 0.9. What’s the distribution of the sum of four Bernoulli random variables?

Thank you, I don't really follow? Sorry I wasn't specific, for c, I dont understand why you use a binomial how you know to?

The distribution of the number of confidence intervals containing mu, out of the four, will be binomial. Whether each confidence interval contains mu is bernoulli with probability 0.9, therefore the distribution of the number of confidence intervals put of the four that contain mu will be binomial with parameters 4 and 0.9.