Matt Gershoff's 100th Episode Puzzler Solution
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Make a copy of this file to play around with the values in it yourself.
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Podcast episode to which this pertains:
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The question posed:
Why might using 1/sqrt(n) be a good rough guess for a 95% confidence interval (CI)?
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We're working with population proportions here (the proportion of our population -- estimated by our sample -- that converts). Search for "confidence interval for a proportion" for details. And then
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consider the 95% CI formula when P = 0.5, as well as about how close 1.96 is to 2.
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The shortcut provides a slightly larger (slightly more conservative) confidence interval, as "clearly" illustrated below (if you concentrate).
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Sample Size (n):24
← Feel free to change
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95.45% CI for a 50% Conversion Rate
0.2041
← The 1/sqrt(n) shortcut proposed by "Tim" (Matt)
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95% CI for a 50% Conversion Rate
0.2000← 1.96 SE
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True 95% CIShortcut: CI Using 1/sqrt(n)
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Sample Conversion Rate (P)
P*(1-P)Standard Deviation (SD)Standard Error (SE)1.96 SE (95% CI)LowerUpperLowerUpper
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5%4.75%21.79%4.45%8.72%-3.7%13.7%-15.4%25.4%
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10%9.00%30.00%6.12%12.00%-2.0%22.0%-10.4%30.4%
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15%12.75%35.71%7.29%14.29%0.7%29.3%-5.4%35.4%
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20%16.00%40.00%8.16%16.00%4.0%36.0%-0.4%40.4%
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25%18.75%43.30%8.84%17.32%7.7%42.3%4.6%45.4%
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30%21.00%45.83%9.35%18.33%11.7%48.3%9.6%50.4%
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35%22.75%47.70%9.74%19.08%15.9%54.1%14.6%55.4%
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40%24.00%48.99%10.00%19.60%20.4%59.6%19.6%60.4%
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45%24.75%49.75%10.16%19.90%25.1%64.9%24.6%65.4%
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50%25.00%50.00%10.21%20.00%30.0%70.0%29.6%70.4%
← This is where P*(1-P), and, consequently, SD and SE, are at their max.
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55%24.75%49.75%10.16%19.90%35.1%74.9%34.6%75.4%
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60%24.00%48.99%10.00%19.60%40.4%79.6%39.6%80.4%
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65%22.75%47.70%9.74%19.08%45.9%84.1%44.6%85.4%
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70%21.00%45.83%9.35%18.33%51.7%88.3%49.6%90.4%
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75%18.75%43.30%8.84%17.32%57.7%92.3%54.6%95.4%
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80%16.00%40.00%8.16%16.00%64.0%96.0%59.6%100.4%
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85%12.75%35.71%7.29%14.29%70.7%99.3%64.6%105.4%
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90%9.00%30.00%6.12%12.00%78.0%102.0%69.6%110.4%
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95%4.75%21.79%4.45%8.72%86.3%103.7%74.6%115.4%
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CI1/sqrt(n)95% CI
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High70.4%70.0%
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Mean50.00%50.00%
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Low29.6%30.0%
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