Learning objectives

Confidence interval estimation for a population mean

You saw at the beginning of Chapter 6 that we can use a sample statistic to make an informed guess, or estimate, of the value of the corresponding population parameter. For example, the sample mean birthweight for the 30 infants in Table 2.5 was 3644.4 g, so you can estimate the population mean birthweight of all infants of whom this sample is representative, also to be about 3644 g,1 plus or minus some (hopefully) small random or sampling error. The obvious questions are:

How small is this ‘plus or minus’ bit?

Can it be quantified?

Can we establish how precise our sample mean birthweight is as an estimate of population mean birthweight?

How close to a population mean can you expect any given sample mean to be?

As you can see these are all essentially the same question, ‘How big an error might we be making when we use the sample mean as an estimate of the population mean?’. This question can be answered with what is known as a confidence interval estimator, which is a numeric expression that quantifies the likely size of the sampling error. But to get a confidence interval we need first to introduce an important concept in statistical inference – the standard error.

The standard error of the mean

Our sample of 30 infants produced a sample mean birthweight of 3644.4 g. You could take a second, different, sample of 30 infants from the same population, and this sample would produce a different value for the sample mean. And a third sample, and a fourth and so on. In fact from any realistic population you could (in theory), take a huge number of different same-size samples, each of which would produce a different sample mean. You would end up with a large number of sample means, and if you were to arrange all of these sample means into a frequency curve, you would find:

That it was Normal. This Normal-ness of the distribution of sample means is a very useful quality (to say the least); we will depend on it a lot in what is to come.

That it was centred around the true population mean. In other words, the mean of all possible sample means is the same as the population mean.

This is very re-assuring. It means that, on average, the sample mean estimates the population mean exactly. But note the ‘on average’. A particular single sample mean may still be some distance from the true mean.

1The value of the sample mean of 3644.4g is known as the point estimate of the population mean. It’s the single best guess you could make as to the value of the population mean.

How we use the standard error of the mean to calculate a confidence interval for a population mean

With the standard error under our belt we can now get to grips with the confidence interval. You have seen that we can be 95 per cent confident that any sample mean is going to be within plus or minus two standard errors of the population mean.3 From this we can show that:

Population mean = sample mean ± 2 × standard error

That is:

We can be 95 per cent confident that the interval, from the sample mean – 2 × standard error, to the sample mean + 2 × standard error, will include the population mean.

Or in probability terms, there is a probability of 0.95 that the interval from the sample mean

– 2 × standard error, to the sample mean + 2 × standard error, will contain the population mean.

In other words, if you pick one out of all the possible sample means at random, there is a probability of 0.95 that it will lie within two standard errors of the population mean. We call the distance from the sample mean – 2 × s.e.(x¯), to the sample mean + 2 × s.e.(x¯), the confidence interval.

The above result means that you now quantify just how close a sample mean is likely to be to the population mean. For obvious reasons the value you get when you put some figures into this expression is known as the 95 per cent confidence interval estimate of the population mean. A 95 per cent confidence level is most common, but 99 per cent confidence intervals are also used on occasion. Note that the confidence interval is sometimes said to represent a plausible range of values for the population parameter.

A worked example from practice

In the cord-platelet count histogram in Figure 4.6, the mean cord platelet count in a sample of 4382 infants is 306×109/l, and the standard deviation is 69×109/l, so the standard error of the mean is:

¯ = × 9 √ = × 9 s.e.(X) 69 10 / 4382 1.042 10 / l

3I have used the value two in all of these expressions as a convenient approximation to the exact value (which in any case will be very close to two, when the probability is 0.95). The exact value comes from what is known as the t distribution. The t distribution is similar to the Normal distribution, but for small sample sizes is slightly wider and flatter. It is used instead of the Normal distribution for reasons connected to inferences about the population standard deviation, which we don’t need to go into here. Anyway, in practice you will use a computer to obtain your confidence interval result. This will use the proper value.

You can see that in the integrated care group of 296 subjects, the sample mean number of bronchodilators prescribed over 12 months was 10.1, with a 95 per cent confidence interval for the population mean of (9.2 to 11.1). So you can be 95 per cent confident that the population mean number of bronchodilators prescribed for this group is somewhere between 9.2 and 11.1. In the control group, the sample mean is 10.6 with a 95 per cent confidence interval for the population mean (9.7 to 11.7), which can be similarly interpreted.

Exercise 9.3 Interpret and compare the sample mean number of hospital admissions, and their corresponding confidence intervals, for the two groups in Table 9.1.

Confidence intervals as described above can also be applied to a population percentage, provided that the values are percentages of a metric variable, for example percentage mortality across a number of hospitals following some procedure (see, for example, Table 2.7). However, if the data is a proportion or percentage of a nominal or ordinal variable, say the proportion of patients with a pressure sore, or the proportion of mothers with an Edinburgh Maternal Depression Scale score of more than 8, then a different approach, described next, is needed.

Confidence interval for a population proportion

We start with an expression for the standard error of the sample proportion:

p(1 − p)

s.e. = ( p)

n

where p is the sample proportion, and n is sample size. Incidentally, the sampling distribution of sample proportions has a binomial distribution, which is quite different from the Normal distribution if the sample is small, but becomes more Normal as sample size increases. The 95 per cent confidence interval for the population proportion is equal to the sample proportion plus or minus 1.96 4 standard errors:

{[ p − 1.96 × s.e.( p)] to [ p + 1.96 × s.e.( p)]}

For example, from Table 1.6, 14 of the 106 women with a malignant diagnosis are premenopausal giving a sample proportion p of 14/106 or 0.13. The standard error of p is thus:

4When we are dealing with proportions, we use, not the t distribution, but the z, or Standard Normal, distribution. The 95 per cent value for z is 1.96.