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.
Find the mean and the variance of \(S\).
Find an approximation for the probability that the mean of the 40 scores is less than 3 (3)
A factory produces steel sheets whose weights \(X \mathrm {~kg}\), are such that \(X \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\)
A random sample of these sheets is taken and a \(95 \%\) confidence interval for \(\mu\) is found to be (29.74, 31.86)
Find, to 2 decimal places, the standard error of the mean.
Hence, or otherwise, find a \(90 \%\) confidence interval for \(\mu\) based on the same sample of sheets.
Using four different random samples, four \(90 \%\) confidence intervals for \(\mu\) are to be found.
Calculate the probability that at least 3 of these intervals will contain \(\mu\).