confidence intervals

Question

A random sample of customer order totals with an average of $78.25 and a population standard deviation of $22.50. Calculate a 90% confidence interval for the mean, given a sample size of 75 orders.

Answer 1

Okay, so first we want to know how many stdevs from the mean will cover 90% of the results.  That number we can get out of a calculator or a chart.  On my chart, I see that a result of 1.64 SDs covers 89.9%, and 1.65 SDs covers 90.1%, so we're looking for halfway between those at 1.644 SDs. (Actually, you should just memorize 1.644 SDs for 90%.)

 

Also, because of the central limit theorem, we know that the larger the sample size N, the smaller the standard deviation by a ratio of sqrt(N).

 

So, our plus-or-minus figure is going to be 1.644 * $22.50 / sqrt(75) = $4.27

 

So, we can have 90% confidence that our actual mean is in the range 78.25 +/1 4.27, meaning between $73.98 and $82.52.

 

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Here's a calculator online you can pop the numbers into, that gives the same result.

 

https://www.mccallum-layton.co.uk/tools/statistic-calculators/confidence-interval-for-mean-calculator/#confidence-interval-for-mean-calculator