Binomial Proportion Confidence Interval - Wilson Score Interval

Wilson Score Interval

The Wilson interval is an improvement (the actual coverage probability is closer to the nominal value) over the normal approximation interval and was first developed by Edwin Bidwell Wilson (1927).

\frac{{ {\hat p + \frac{{1}}{{2n}} z_{1- \alpha / 2}^2 \pm z_{1- \alpha / 2} \sqrt {\frac{{\hat p\left( {1 - \hat p} \right)}}{n} + \frac{{z_{1- \alpha / 2}^2}} {{4n^2}} }} }} {{ {1 + \frac{{1}}{n}} z_{1- \alpha / 2}^2 }}

This interval has good properties even for a small number of trials and/or an extreme probability. The center of the Wilson interval

\frac{{ {\hat p + \frac{{1}}{{2n}} z_{1- \alpha / 2}^2 } }} {{ {1 + \frac{{1}}{n}} z_{1- \alpha / 2}^2 }}

can be shown to be a weighted average of and, with receiving greater weight as the sample size increases. For the 95% interval, the Wilson interval is nearly identical to the normal approximation interval using instead of .

The Wilson interval can be derived from Pearson's chi-squared test with two categories. The resulting interval

can then be solved for to produce the Wilson interval.

The test in the middle of the inequality is a score test, so the Wilson interval is sometimes called the Wilson score interval.

Read more about this topic:  Binomial Proportion Confidence Interval

Famous quotes containing the words wilson, score and/or interval:

    “With proper attribution, to quote another’s thoughts and words is appropriate; plagiarism, however, is cheating, and it may break copyright law as well.”
    —Kenneth G. Wilson (1923)

    “A man who is a politician at forty is a statesman at three score and ten. It is at this age, when he would be too old to be a clerk or a gardener or a police-court magistrate, that he is ripe to govern a country.”
    —W. Somerset Maugham (1874–1965)

    “I was interested to see how a pioneer lived on this side of the country. His life is in some respects more adventurous than that of his brother in the West; for he contends with winter as well as the wilderness, and there is a greater interval of time at least between him and the army which is to follow. Here immigration is a tide which may ebb when it has swept away the pines; there it is not a tide, but an inundation, and roads and other improvements come steadily rushing after.”
    —Henry David Thoreau (1817–1862)