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 $$\bar{x}$$, the sample mean (the statistic). The actual value of $$\mu$$ might be a bit larger than $$\bar{x}$$, or a bit smaller than $$\bar{x}$$; that is, $$\mu$$ is probably about $$\bar{x}$$, give-or-take a bit.

Furthermore, we have seen that the values of $$\bar{x}$$ 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 $$\bar{x}$$ give-or-take about two standard errors. In general, the confidence interval (CI) for $$\mu$$ is

$\bar{x} \pm \overbrace{(\text{Multiplier}\times\text{s.e.}(\bar{x}))}^{\text{Called the `margin of error'}}.$ 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, $$\bar{x}$$ 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 $$\bar{x}$$.

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 $$\bar{x} = 2.8$$ kg with a standard deviation of $$s=0.94$$ kg .

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

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

$2.8\pm(2 \times 0.03883),$ or $$2.8\pm0.07766$$. (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.

References

Dianat I, Sorkhi N, Pourhossein A, Alipour A, Asghari-Jafarabadi M. Neck, shoulder and low back pain in secondary schoolchildren in relation to schoolbag carriage: Should the recommended weight limits be gender-specific? Applied Ergonomics. 2014;45:437–42.