Question

question
tyler is a student in an environmental science class looking into the seasonal snowfall totals for two nearby locations. one location is on the coastline, while a second location is 30 miles inland. he randomly selects 10 seasons for the coastal location and another 10 seasons for the inland location, recording the annual snowfall total in inches for each season. the results of tyler’s survey are shown in the samples provided. both locations have a season that had 48.6 in. of total snowfall.
based on the ( z ) scores calculated above, is the 48.6 in. snowfall amount more unusual for the coastal location or the inland location?

coastal	inland
31.7	51.3
34.8	48.6
41.1	58.6
38.2	62.7
36.9	58
48.6	59.3
43.4	51.6
40.5	65.4

select the correct answer below:

• the 48.6 inches of snowfall at the coastal location is more unusual because the absolute value of the ( z )-score for the coastal location is less than the absolute value of the ( z )-score for the inland location.
• the 48.6 inches of snowfall at the coastal location is more unusual because the absolute value of the ( z )-score for the coastal location is greater than the absolute value of the ( z )-score for the inland location.
• the 48.6 inches of snowfall at the inland location is more unusual because the absolute value of the ( z )-score for the inland location is greater than the absolute value of the ( z )-score for the coastal location.
• the 48.6 inches of snowfall at the inland location is more unusual because the absolute value of the ( z )-score for the inland location is less than the absolute value of the ( z )-score for the coastal location.

Explanation:

Step1: Recall z - score concept

The z - score formula is \(z=\frac{x - \mu}{\sigma}\), where \(x\) is the data point, \(\mu\) is the mean, and \(\sigma\) is the standard deviation. The more unusual a data point is, the larger the absolute value of its z - score (since a larger absolute z - score means the data point is further from the mean in terms of standard deviations).

Step2: Analyze the relationship between z - score and unusualness

For a given data point \(x = 48.6\), we compare the absolute values of the z - scores for the coastal and inland locations. If the absolute value of the z - score for the coastal location is greater than that for the inland location, then 48.6 is more unusual for the coastal location. If the absolute value of the z - score for the inland location is greater, then it is more unusual for the inland location.

From the options, the correct reasoning is: The 48.6 inches of snowfall at the coastal location is more unusual because the absolute value of the z - score for the coastal location is greater than the absolute value of the z - score for the inland location.

Answer:

The 48.6 inches of snowfall at the coastal location is more unusual because the absolute value of the \(z\) - score for the coastal location is greater than the absolute value of the \(z\) - score for the inland location. (The option corresponding to this statement)