Question

a towns january high temperatures average 38°f with a standard deviation of 9°f, while in july the mean high temperature is 72°f and the standard deviation is 9°f. in which month is it more unusual to have a day with a high temperature of 57°f? explain.
select the correct choice below and fill in the answer boxes within your choice.
(round to three decimal places as needed.)
a. it is more unusual to have a day with a high temperature of 57°f in july. a high temperature of 57°f in july is standard deviations below the mean and a high temperature of 57°f in january is only standard deviations below the mean.
b. it is more unusual to have a day with a high temperature of 57°f in january. a high temperature of 57°f in january is standard deviations above the mean and a high temperature of 57°f in july is only standard deviations below the mean.

Explanation:

Step1: Calculate z - score for January

The z - score formula is $z=\frac{x - \mu}{\sigma}$, where $x$ is the value, $\mu$ is the mean, and $\sigma$ is the standard deviation. For January, $\mu = 38$, $\sigma=9$, and $x = 57$. So $z_{January}=\frac{57 - 38}{9}=\frac{19}{9}\approx2.11$.

Step2: Analyze the z - score for January

A z - score of approximately 2.11 means the value is 2.11 standard deviations above the mean. In general, values with a z - score greater than 2 (or less than - 2) are considered unusual.

Step3: Calculate z - score for July (assuming a typical July mean is much higher than 57, say 72 is a common mean for July and standard deviation is also 9 for comparison purposes)

If $\mu = 72$, $\sigma = 9$, and $x = 57$, then $z_{July}=\frac{57 - 72}{9}=\frac{- 15}{9}\approx - 1.67$.

Step4: Analyze the z - score for July

A z - score of approximately - 1.67 means the value is 1.67 standard deviations below the mean. Since $| - 1.67|<2$, it is not considered unusual.

Answer:

It is more unusual to have a day with a high temperature of $57^{\circ}$ in January because the z - score for 57 in January is approximately 2.11 (more than 2 standard deviations above the mean), while the z - score for 57 in July (assuming typical July mean) is approximately - 1.67 (less than 2 standard deviations from the mean).