Chance the average temperature lands on target

Question

A sensor's temperature reading has a true mean of $100^\circ$ C with standard deviation $\sigma = 4^\circ$ C. Readings are independent, so by the central limit theorem the average of many readings is approximately normal.

Over $n = 64$ readings, what is the probability the average lands within $1^\circ$ C of $100^\circ$ C (that is, between $99$ and $101$ )?

Accepted Answer

This is a two-sided version: instead of one tail, we want the probability of landing in a central band. The average of 64 readings is approximately normal, centered at 100^\circC, with spread equal to the standard error: 	ext{SE} = \frac{\sigma}{\sqrt{n}} = \frac{4}{\sqrt{64}} = \frac{4}{8} = 0.5^\circ	ext{C}. The band 99 to 101 is \pm 1^\circC around the mean. In standard-error units that is z = \frac{1}{0.5} = 2, so we want the probability of being within \pm 2 standard errors: P(-2 < Z < 2) \approx 0.9545, about 95.5\%. This is just the familiar "2 sigma covers about 95\%" rule, applied …

Chance the average temperature lands on target

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