9.4: Confidence Intervals for Means

What if σ is unknown?

As we discussed earlier, when variables have been well-researched in different populations it is reasonable to assume that the population standard deviation (σ) is known. However, this is rarely the case. What if σ is unknown?

Well, there is some good news and some bad news.

The good news is that we can easily replace the population standard deviation, σ, with the sample standard deviation, s.

The bad news is that once σ has been replaced by s, we lose the Central Limit Theorem, together with the normality of $\overline{X}$, and therefore the confidence multipliers z* for the different levels of confidence (1.645, 2, 2.576) are (generally) not accurate any more. The new multipliers come from a different distribution called the “t distribution” and are therefore denoted by t* (instead of z*). We will discuss the t distribution in more detail when we talk about hypothesis testing.

The confidence interval for the population mean (μ) when (σ) is unknown is therefore:

[latex]\bar{\mathcal{x}}\pm\mathcal{t}^**\frac{\mathcal{s}}{\sqrt\mathcal{n}}[/latex]

(Note that this interval is very similar to the one when σ is known, with the obvious changes: s replaces σ, and t* replaces z* as discussed above.)

There is an important difference between the confidence multipliers we have used so far (z*) and those needed for the case when σ is unknown (t*). Unlike the confidence multipliers we have used so far (z*), which depend only on the level of confidence, the new multipliers (t*) have the added complexity that they depend on both the level of confidence and on the sample size (for example, the t* used in a 95% confidence when n = 10 is different from the t* used when n = 40). Due to this added complexity in determining the appropriate t*, we will rely heavily on software in this case.

Comments

1. Since it is quite rare that σ is known, this interval (sometimes called a one-sample t confidence interval) is more commonly used as the confidence interval for estimating μ. (Nevertheless, we could not have presented it without our extended discussion up to this point, which also provided you with a solid understanding of confidence intervals.)
2. The quantity [latex]\frac{\mathcal{s}}{\sqrt\mathcal{n}}[/latex] is called the standard error of $\overline{X}$. The central limit theorem tells us that [latex]\frac{\sigma}{\sqrt\mathcal{n}}[/latex] is the standard deviation of $\overline{X}$ (and this is the quantity used in confidence interval when σ is known). In general, whenever we replace parameters with their sample counterparts in the standard deviation of a statistic, the resulting quantity is called the standard error of the statistic. In this case, we replaced σ with its sample counterpart (s), and thus [latex]\frac{\mathcal{s}}{\sqrt\mathcal{n}}[/latex] is the standard error of (the statistic) $\overline{X}$.
3. As before, to safely use this confidence interval, the sample must be random, and the only case when this interval cannot be used is when the sample size is small and the variable is not known to vary normally.

Final comment

It turns out that for large values of n, the t* multipliers are not that different from the z* multipliers, and therefore using the interval formula:

[latex]\bar{\mathcal{x}}\pm\mathcal{z}^**\frac{\mathcal{s}}{\sqrt\mathcal{n}}[/latex]

for μ when σ is unknown provides a pretty good approximation.

Let’s summarize

* When the population is normal and/or the sample is large, a confidence interval for unknown population mean μ when σ is known is:

* There is a trade-off between the level of confidence and the precision of the interval estimation. The price we have to pay for more precision is sacrificing level of confidence.

* The general form of confidence intervals is an estimate +/- the margin of error (m). In this case, the estimate=$\overline{x}$ and [latex]\mathcal{m}=\mathcal{z}\ast\frac{\sigma}{\sqrt\mathcal{n}}[/latex]. The confidence interval is therefore centered at the estimate and its width is exactly 2m.

* For a given level of confidence, the width of the interval depends on the sample size. We can therefore do a sample size calculation to figure out what sample size is needed in order to get a confidence interval with a desired margin of error m, and a certain level of confidence (assuming we have some flexibility with the sample size). To do the sample size calculation we use:

[latex]\mathcal{n}=\left(\frac{\mathcal{z}^*\sigma}{\mathcal{m}}\right)^2[/latex]

(and round up to the next integer).

* When σ is unknown, we use the sample standard deviation, s, instead, but as a result we also need to use a different set of confidence multipliers (t*) associated with the t distribution. The interval is therefore

[latex]\bar{\mathcal{x}}\pm\mathcal{t}^**\frac{\mathcal{s}}{\sqrt\mathcal{n}}[latex]

* These new multipliers have the added complexity that they depend not only on the level of confidence, but also on the sample size. Software is therefore very useful for calculating confidence intervals in this case.

* For large values of n, the t* multipliers are not that different from the z* multipliers, and therefore using the interval formula:

[latex]\bar{\mathcal{x}}\pm\mathcal{z}^**\frac{\mathcal{s}}{\sqrt\mathcal{n}}[/latex]

for μ when σ is unknown provides a pretty good approximation.