Answer:
Explanation:
A z-score represents how many standard deviations an observation is away from the mean of a distribution. The formula for calculating a z-score is:
z = (x - μ) / σ
Where x is the observation, μ is the mean of the distribution, and σ is the standard deviation of the distribution.
For example, if we have a distribution of high temperatures in January with a mean of 37°F and a standard deviation of 12°F, and we observe a high temperature of 56°F, we can calculate the z-score as follows:
z = (56 - 37) / 12 = 1.58
This means that a high temperature of 56°F in January is 1.58 standard deviations above the mean. A high z-score indicates that the observation is relatively far away from the mean of the distribution, and therefore is more unusual.
Now, let's consider the same high temperature of 56°F, but this time in July, where the mean high temperature is 75°F and the standard deviation is 6°F. The z-score for this observation would be:
z = (56 - 75) / 6 = -3.17
This means that a high temperature of 56°F in July is 3.17 standard deviations below the mean. This is a much higher absolute value of the z-score than in January, indicating that the observation is even further away from the mean, and therefore even more unusual.
The reason for this is that the standard deviation in July is smaller than in January, which means that temperatures far away from the mean are more unusual. Additionally, the mean in July is higher than in January, so temperatures that are much lower than the mean are more unusual.
Therefore, we can conclude that it is more unusual to have a day with a high temperature of 56°F in July than in January, based on the z-scores and the characteristics of the distributions in each month.