Question

Headaches and Handedness A study was conducted to investigate the relationship between severe headaches and being left- or right-handed. 48 (Incidentally, Lisa Kudrow, who played Phoebe Buffay on the hit sitcom "Friends," is an author on this study.) Of 273 participants with cluster headaches, 24 were left-handed. Of 477 participants with migraine headaches, 42 were left-handed. (a) Give an estimate for the proportion of cluster headache sufferers who are left-handed. (b) Use StatKey or other technology to construct and interpret a \(95 \%\) confidence interval for the proportion of cluster headache sufferers who are left-handed. (c) Give an estimate for the proportion of migraine sufferers who are left- handed. (d) Use StatKey or other technology to construct and interpret a \(95 \%\) confidence interval for the proportion of migraine sufferers who are lefthanded. (e) Compare your confidence intervals in parts (b) and (d). Which is more narrow? Explain why.

Short answer

The estimated proportions of left-handed sufferers are \(0.088\) for both cluster and migraine headaches. The \(95\%\) confidence interval for the proportion of left-handed cluster headache sufferers is approximately \(0.057\) to \(0.119\), while that for left-handed migraine sufferers is approximately \(0.063\) to \(0.113\). So, the confidence interval for migraine sufferers is a bit narrower, likely due to the larger sample size.

Step-by-step solution

Calculate Proportions

In order to estimate the proportions for each type of headache, the number of left-handed sufferers is divided by the total number of sufferers. For cluster headaches, the proportion who are left-handed is calculated as \(24 / 273 = 0.088\). For migraine sufferers, the estimated proportion is calculated as \(42 / 477 = 0.088\).

Construct the Confidence Interval for Cluster Headaches

Using a technology like StatKey, you will generate a \(95\%\) confidence interval for the population proportion. For our calculation, we find the interval to be approximately \(0.057\) to \(0.119\). This suggests with \(95\%\) confidence that the true proportion of left-handed cluster headache sufferers is between these two values.

Construct the Confidence Interval for Migraine Headaches

Again StatKey is used to create a \(95\%\) confidence interval for the population proportion of migraine sufferers who are left-handed. The confidence interval is approximately \(0.063\) to \(0.113\). This suggests that the true proportion of left-handed migraine sufferers is between these values with \(95\%\) confidence.

Comparing Confidence Intervals

Looking at both confidence intervals, we see that the interval for migraine sufferers is slightly narrower. This is likely due to the larger sample size for migraine sufferers, which typically leads to smaller margins of error and thus, narrower confidence intervals.