Confidence Interval for Population Mean

Definition of the Confidence Interval for a Population Mean

Remember that confidence intervals are one type of statistical inference. They allow you to find a range of values where you can be relatively confident the true value will be. In this case, you are constructing a confidence interval for a population mean.

The general form for a confidence interval estimate for a population mean is

sample mean \(\pm\) critical value \(\times\) standard error of the statistic.

Here the sample mean, \(\bar{x}\), is an unbiased estimator for the population mean \(\mu\) as long as the sample size \(n\) is such that \(n > 30\) so you can apply the Central Limit Theorem. Remember that being able to use the Central Limit Theorem implies you can assume that your sample is approximately normal, even if the population itself is not.

Let's start by looking at the case where you know the population standard deviation, \(\sigma\). In general, you won't know this, but it is a helpful starting case to go over. Then the standard deviation of the sample is :

\[ \sigma_{\bar{x}} = \frac{\sigma}{n} \]

So when the sample size is large enough, and \(\sigma\) is known, the confidence interval is given by:

\[ \bar{x} \pm (z \text{ critical value})\left(\frac{\sigma}{\sqrt{n}}\right)\]

What about when you don't have the population standard deviation?

Construct the Confidence Interval for the Population Mean

First, let's look at the conditions that need to be satisfied before you can construct the confidence interval for a population mean when you don't know what the population standard deviation is.

• Either the sample size is large enough (\(n\ge 30\) ) or the population distribution is approximately normal.

• The sample is random or it is reasonable to assume it is representative of the larger population.

When these conditions are met, you can construct the confidence interval for the population mean. Unlike the case where you know the standard deviation of the population and can use the \(z\)-value as your critical value, when you don't know the standard deviation of the population you will need to use the \(t\)-distribution instead.

Confidence Interval for Population Mean Formula

When the conditions to construct a confidence interval for the population mean are satisfied, the formula for the confidence interval becomes

\[ \bar{x} \pm (t \text{ critical value})\left(\frac{s}{\sqrt{n}}\right)\]

Here \(s\) is the sample standard deviation. The \(t\) critical value is based on the degree of freedom, \(df\), which is calculated by:

\[df = n-1\]

and the confidence level you are using.

Calculate Confidence Interval for Population Mean

Let's go back to the coffee shop example at the start of the article. The sample mean is \(\bar{x} = 14.5\) minutes, the sample size is \(n=40\), and the sample standard deviation is \(s=1.7\) . The sample size is large enough to construct a confidence interval, and it is reasonable to assume that the coffee drive-through and car samples are random.

Suppose you want to construct a \(95\%\) confidence interval. The degree of freedom is:

\[ df = n- 1 = 39\]

If you are using a \(t\)-table, you won't find \(39\) degrees of freedom on the table! That is because for any degree of freedom larger than \(30\) the increase isn't very much when you just increase the number of degrees of freedom by \(1\).

That is why you will see tables go by \(1\) up to \(30\), and then start to jump by \(10\). The difference between the \(t\)-critical value for \(df = 39\) and \(df = 40\) isn't really big enough to affect your calculations, so you can use the table value for \(df = 40\) and a \(95\%\) confidence interval instead. There you will find that the appropriate \(t\)-critical value is \(2.02\).

Then the confidence interval for the population mean is:

\[\begin{align} \bar{x} \pm (t \text{ critical value})\left(\frac{s}{\sqrt{n}}\right) &= 14.5 \pm (2.02)\left( \frac{1.7}{\sqrt{40}}\right) \\ &\approx 14.5 \pm 0.54 \\ &= (13.96,15.04) \end{align}\]

You still need to interpret the confidence interval and communicate your findings to other people. You can say two things:

• The method used to construct the confidence interval estimate will capture the actual population mean \(95\%\) of the time.

• You are \(95\%\) confident that the actual mean time it takes to get coffee in a drive-through is between \(13.96\) and \(15.04\) minutes.

Examples of Confidence Intervals for Population Means

You may need to determine how many samples you need in advance to make sure the margin of error is relatively small. This can be important when gathering data is expensive or time-consuming. Remember that the margin of error is defined as the maximum likely estimation error you can expect when using the statistic is used as an estimator.

The margin of error is given by the formula:

margin of error = \(1.96 \left(\dfrac{\sigma}{\sqrt{n}}\right)\).

When you don't know the population standard deviation, you can approximate the margin of error using the formula

estimated margin of error = \(1.96 \left(\dfrac{s}{\sqrt{n}}\right)\).

If you let \(M\) be the margin of error, solving for the sample size \(n\) gives you:

\[ n = \left( \frac{1.96 \sigma}{M} \right)^2 \]

Sometimes you won't have even a sample standard deviation. In that case, a rough estimate can be found by taking:

\[\begin{align} \sigma &\approx \frac{\text{largest sample value} - \text{smallest sample value} }{4} \\ &= \frac{\text{range}}{4} \end{align}\]

as long as the data isn't too skewed.

Let's look at an example where you need to use the estimated margin of error to find a sample size.

Let's take a look at another example.

Now let's construct the confidence interval for the price of coffee.