Question

Thermometer Error. Assume that the readings at freezing on a batch of thermometers are normally distributed with a mean of 0.0°C and a standard deviation of 1.0°C. What readings separates the highest 13% of the thermometers from the rest? (Round answer to two decimal places)

Answer 1

A reading of 1.25ºC separates the highest 13% of the thermometers from the rest.

Explanation:

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

Assume that the readings at freezing on a batch of thermometers are normally distributed with a mean of 0.0°C and a standard deviation of 1.0°C. This means that
`\mu = 0, \sigma = 1`.

What readings separates the highest 13% of the thermometers from the rest?

This is the value of X when Z has a pvalue of 0.87.

This is
`Z = 1.25`. So

`Z = (X - \mu)/(\sigma)`

`1.25 = (X - 0)/(1)`

`X = 1.25`

A reading of 1.25ºC separates the highest 13% of the thermometers from the rest.