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I am currently reading Math behind A/B testing written by Amazon and got stuck. At some point they say:

To determine the 95% confidence interval on each side of conversion rate, we multiply the standard error with the 95th percentile (one tailed) of a standard normal distribution (a constant value equal to 1.65).

Then they use that constant to calculate the confidence interval:

range = conversion rate +- (1.65 x Standard Error)

I read somewhere to get the aforementioned constant value from the following table:

http://www.sjsu.edu/faculty/gerstman/StatPrimer/t-table.pdf enter image description here

The problem is that I can't see 1.65 anywhere for 95% and the closest value is 1.960, hence my confusion.

Could someone explain me where the 1.65 is coming from?

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I think it's a mistake. For a two-sided confidence interval the two-sided test is appropriate - for a 95% interval your value of 1.96 is correct. The one-sided value (1.65) would be appropriate only if you wanted to calculate a confidence region from $-\infty$ to $c$ or from $c$ to $\infty$.

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The value 1.65 belongs to the 90% confidence interval. You can find it (actually 1.645 in the table) in the table above in the bottom row and the column labeled (at the bottom) 90%. For a 95% confidence interval the constant would be 1.96 (the value next to it).

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  • $\begingroup$ That's what I thought as well. $\endgroup$ – Max Apr 16 '13 at 11:36
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It's the rounded value of 1.645, found in the Z row (corresponding to inf number of degrees of freedom and therefore normal distribution) under the t.95 column.

95th percentile of one-tailed distribution is the distance in standard deviations from the mean to the point in either one end of the axis that separates the values in one region (1 tail) having total probability of 5% and the remainder with 95%. For a two-tailed version that same distance (1.645) would result in two same regions at both ends of the axis, summing to 10% and leaving 90% in the middle. Hence one-tail 0.05 and two-tail 0.10 in the column name.

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