Question

a set of average city temperatures in december are normally distributed with a mean of 16.3°C and a standard deviation of 2°C. what proportion of temperatures are between 12.9°C and 14.9°C?

Answer 1

Answer:

19.74% of temperatures are between 12.9°C and 14.9°C

Step-by-step explanation:

Problems of normally distributed samples are 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:

$Z = \frac{X - \mu}{\sigma}$

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 question, we have that:

$\mu = 16.3, \sigma = 2$

What proportion of temperatures are between 12.9°C and 14.9°C?

This is the pvalue of Z when X = 14.9 subtracted by the pvalue of Z when X = 12.9.

X = 14.9

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{14.9 - 16.3}{2}$

$Z = -0.7$

$Z = -0.7$ has a pvalue of 0.2420

X = 12.9

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{12.9 - 16.3}{2}$

$Z = -1.7$

$Z = -1.7$ has a pvalue of 0.0446

0.2420 - 0.0446 = 0.1974

19.74% of temperatures are between 12.9°C and 14.9°C