22.3 One mean: Confidence intervals

We don’t know the value of $\mu$ (the paremeter), the population mean, but we have an estimate: the value of $\overline{x}$, the sample mean (the statistic). The actual value of $\mu$ might be a bit larger than $\overline{x}$, or a bit smaller than $\overline{x}$; that is, $\mu$ is probably about $\overline{x}$, give-or-take a bit.

Furthermore, we have seen that the values of $\overline{x}$ vary from sample to sample (sampling variation), and noted that they vary with an approximate normal distribution. So, using the 68–95–99.7 rule, we could create an approximate 95% interval for the plausible values of $\mu$ that may have given the observed values of the sample mean. This is a confidence interval.

A confidence interval (CI) for the population mean is an interval surrounding a sample mean. In general, an approximate 95% confidence interval (CI) for $\mu$ is $\overline{x}$ give-or-take about two standard errors. In general, the confidence interval (CI) for $\mu$ is

$$\overline{x}\pm \stackrel{\text{Called the `margin of error'}}{\stackrel{}{(\text{Multiplier}\times \text{s.e.}(\overline{x}\left)\right)}}.$$ For an approximate 95% CI, the multiplier is, as usual, about $2$ (since about 95% of values are within two standard deviations of the mean from the 68–95–99.7 rule).

We often find 95% CIs, but we can find a CI with any level of confidence: we just need a different multiplier. We’ll just use a multiplier of $2$ (and hence find approximate 95% CIs), and otherwise use software. Commonly, CIs are computed at 90%, 95% and 99% confidence levels.

The multiplier of 2 is not a $z$-score here. The multiplier would be a $z$-score if we knew the value of $\sigma$; since we don’t, the multiplier is a $t$-score and not a $z$-score The $t$- and $z$-multipliers are very similar, and (except for very small sample sizes) using an approximate multiplier of 2 is reasonable for computing approximate 95% CIs in either case. We’ll let software handle the specifics.

If we collected many samples of a specific size, $\overline{x}$ and $s$ would be different for each sample, so the calculated CI would be different for each. Some CIs would straddle the population mean $\mu$, and some would not; and we never know if the CI computed from our single sample straddles $\mu$ or not.

Loosely speaking, there is a 95% chance that our 95% CI straddles $\mu$. For a CI computed from a single sample, we don’t know if our CI includes the value of $\mu$ or not. The CI could also be interpreted as the range of plausible values of $\mu$ that could have produced the observed value of $\overline{x}$.

Example 22.1 (School bags) A study of the school bags that 586 children (in Grades 6–8 in Tabriz, Iran) take to school found that the mean weight was $\overline{x}=2.8$ kg with a standard deviation of $s=0.94$ kg (Dianat et al. 2014).

The parameter is the population mean weight of school bags for Iranian children in Grades 6–8.

Of course, another sample of 586 children would produce a different sample mean: the sample mean varies from sample to sample.

The standard error of the sample mean is

$$\text{s.e.}\left(\overline{x}\right)=\frac{s}{\sqrt{n}}=\frac{0.94}{\sqrt{586}}=0.03883;$$ see Fig. 22.1. The approximate 95% CI for the population mean school-bag weight is

$$2.8\pm (2\times 0.03883),$$ or $2.8\pm 0.07766$. (The margin of error is 0.07766.) This is equivalent to an approximate 95% CI from 2.72 kg to 2.88 kg. This CI has a 95% chance of straddling the population mean bag weight.

Think 22.1 (Width of CI) Would a 99% CI for $\mu$ be wider or narrower than the 95% CI? Why?

A wider interval is needed to be more confident that the interval contains the population mean.

FIGURE 22.1: The normal distribution, showing how the sample mean bag weight varies in samples of size $n=586$