Confidence Intervals

We mentioned already that an often chosen critical p value is 0.05. On a two-tailed test, a Z-score of 2 standard errors (SE) corresponds to a p value of 0.046, so 2 SE is a good approximation to threshold for significance. In other words, without performing any statistical calculation, if a sample mean is quoted with its SE (e.g. the sample of systolic blood pressures had a mean (±SE) of 185 mmHg ± 5 mmHg, we can see if it is unlikely to match up to the population mean simply by “eyeballing” the figures or looking at a chart with “error bars” above and below the mean value and seeing if the two means are separated by more than 2 SE.

A more formal way of doing this is by quoting not the SE but the confidence interval (CI). This is the exact range for a given critical p-value. Here we can make a direct comparison to see if the population mean falls within the confidence interval range of the sample mean. In a two-tailed test with a critical p value of 0.05, we have a range corresponding to the p=0.025 level in either direction and from the table the 95% confidence interval range is more accurately 1.96 SE in either direction from the mean.

For a one-tailed test, we would have a range corresponding to the p=0.05 level in one direction. Here, the CI would be 1.64 SE in one direction from the mean. In our one-tailed example of blood pressure, this equates to 8.2 mmHg above and below the mean. We probably shouldn’t use the ± sign, but instead quote a sample mean (- CI) of 185 mmHg – 8.2 mmHg; we would see immediately that this was consistent with the desired bp of 180 mmHg. We could of course turn it around and express the desired mean level as 180 mmHg + 8.2 mmHg, and see that the sample mean of 185 mmHg fell within this range. The left (lower) tail of the sample mean corresponds to the right (upper) tail of the desired mean.

If the hypothesis instead reflected concern that the blood pressures in the company were too low not too high, e.g. if the chief medical officer was only concerned with risk of fainting, this is also a one-tailed test but with focus on the other tail. We would quote the same sample as 185 mmHg + 8.2 mmHg, and see even more obviously that even the mean, let alone this upper limit, is well within the desired minimum 180 mmHg level.

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