Question

Assume that the readings on the thermometers are normally distributed with a mean of 0 degrees and standard deviation of 1.00°C. A thermometer is randomly selected and tested. Draw a sketch and find the temperature reading corresponding to Upper P 84​, the 84 th percentile. This is the temperature reading separating the bottom 84 % from the top 16 %. The temperature for P 84 is approximately_________?

Answer 1

The temperature for P 84 is approximately 1ºC.

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 on the thermometers are normally distributed with a mean of 0 degrees and standard deviation of 1.00°C. This means that
`\mu = 0` and
`\sigma = 1`

This is the temperature reading separating the bottom 84 % from the top 16 %.

This temperature is the value of X when Z has a pvalue of 0.84. This happens at
`Z = 1`.

So

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

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

`X = 1`

The temperature for P 84 is approximately 1ºC.