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Solution to question 4.2

a) True

If we want to be more confident (95% confident rather than 90% confident) that our range of plausible values includes the true value then it makes sense that the inteval has to be wider.

b) False

This statement is confusing standard deviations and standard errors. We would expect 95% of a normally distributed population to lie within the interval

'population mean - 1·96 standard deviations' to 'population mean + 1·96 standard deviations'

A confidence interval is an expression of how confident we are about the value of a single quantity (such as the mean) not a description of a population or sample.

c) Just about true

In loose language there is nothing too wrong about about this statement; but statisticians don't like it. The reason for this is it seems to imply that the mean that could have any of the possible values within the confidence interval. This isn't the case: the mean only has one value, although we don't know it. A preferred statement would be "the 95% confidence interval has a 95% chance of including the mean".

Another way to think of it is to consider repeating the experiment. We would generally get a slightly different 95% confidence interval to the first experiment. The confidence interval changes but, obviously, the underlying (unknown) mean does not. Consequently, we prefer to make sure that any words that imply probability or variability are clearly attached to the words 'confidence interval' and not 'mean'.

d) Frequently true

This is certainly true for the confidence intervals about the mean and the other confidence intervals you will calculate in these exercises. As the course progresses we shall come across non-symmetric confidence intervals.

e) True

This is a good way to express the idea of a 95% confidence interval.

f) True

An absolutely accurate description of the way confidence intervals work.

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