← Use statistical methods to make a formal inference

4 Reading the confidence interval: the make-or-break bit

The three cases

• Interval entirely above zero (e.g. 2.1 to 8.4). You can call it: group A’s median is greater than group B’s back in the populations.
• Interval entirely below zero (e.g. −9.2 to −1.5): the reverse call.
• Interval contains zero (e.g. −1.3 to 4.8). You cannot make the call. Zero is a plausible difference, so the data are consistent with no difference at all.

Worked wording Making the call when the interval excludes zero Bootstrap 90% CI for (median height of boys − median height of girls) = 4.2 cm to 11.6 cm. State the interval: “I am 90% confident that the difference between the median height of all Year 13 boys and all Year 13 girls is between 4.2 cm and 11.6 cm.” Note that the whole interval is above zero. Make the formal inference: “Because the entire interval lies above zero, I can make the call that the median height of all Year 13 boys is greater than that of all Year 13 girls.” Add the practical size: “The difference is somewhere between about 4 and 12 cm. A difference large enough to be noticeable in practice.” ✓ Interval excludes zero ⟹ make the call, and say by roughly how much. Worked wording When the interval INCLUDES zero Bootstrap 90% CI for the difference between medians = −1.3 to 4.8 marks. “I am 90% confident the difference between the population medians lies between −1.3 and 4.8 marks.” “Because this interval contains zero, a difference of zero is plausible.” Inference: “I cannot make the call that one population median is greater than the other.” ⚠️ Do NOT write “there is no difference”, that overstates it. The correct wording is that you cannot make the call. ✓ Interval includes zero ⟹ no call. “Cannot make the call”, not “no difference”. ✕

What “90% confident” does NOT mean

It does not mean there is a 90% chance the true difference is in this interval. It means: if you repeated the whole sampling-and-bootstrapping process many times, about 90% of the intervals you built would capture the true population difference. Examiners notice the difference in wording.

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