Question

The mean high temperature on a particular day in January is 31 degrees F and the standard deviation is 8.7 degrees. One year, the temperature was 52 degrees F on that day. What is the Z-score for that day's temperature? Round to the appropriate number of decimal places for Z-scores. Is this temperature significantly high? (greater than 2)​

Answer 1

Answer:

The Z-score for that day's temperature is 2.41, and since Z > 2, this temperature is significantly high.

Step-by-step explanation:

Z-score:

In a set with mean $\mu$ and standard deviation $\sigma$, the z-score 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 p-value, we get the probability that the value of the measure is greater than X.

If $|Z| > 2$, the measure X is significantly high(Z > 2) or significantly low(Z < -2).

The mean high temperature on a particular day in January is 31 degrees F and the standard deviation is 8.7 degrees.

This means that $\mu = 31, \sigma = 8.7$

One year, the temperature was 52 degrees F on that day.

This means that $X = 52$.

What is the Z-score for that day's temperature?

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

$Z = \frac{52 - 31}{8.7}$

$Z = 2.41$

The Z-score for that day's temperature is 2.41, and since Z > 2, this temperature is significantly high.