Question

Assume that Michigan January daily low temperatures are normally distributed with a mean of 13 F and a standard deviation of 15 F. What is the probability of a January day having a temperature of -20 F?

Answer 1

Answer:

1.39% probability of a January day having a temperature of -20 F

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 problem, we have that:

$\mu = 13, \sigma = 15$

What is the probability of a January day having a temperature of -20 F?

This is the pvalue of Z when X = -20. So

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

$Z = \frac{-20 - 13}{15}$

$Z = -2.2$

$Z = -2.2$ has a pvalue of 0.0139.

1.39% probability of a January day having a temperature of -20 F