Question

describe and correct the error made in describing the association of the following data table.
the data on the table show
positive association because
y is increasing.
choose the correct answer below.
a. in the table, x decreases from left to right. data with x - values that decrease from left to right on a table show negative association.
b. as x increases, y decreases. this means the data in the table show negative association.
c. in the table, y increases from left to right. however, adjacent y - values do not always differ by the same amount. this means the data in the table show no association.
d. one x - value, 17, occurs twice and corresponds to two different y - values, 7 and 8. this means the data in the table show no association.
e. in the table, y increases from left to right. data with y - values that increase from left to right on a table show negative association.

Explanation:

Step1: Analyze option A

In the table, \(x\) values are \(19,18,17,17,15,13,11\). From left - to - right (from \(x = 19\) to \(x = 11\)), \(x\) decreases. \(y\) values are \(3,6,7,8,10,11,12\). From left - to - right, \(y\) increases. So, as \(x\) decreases, \(y\) increases, which shows a negative association. Option A is correct.

Step2: Analyze option B

As \(x\) increases (but in our table \(x\) is decreasing from left - to - right), if we consider the general trend (reverse the order of \(x\) values conceptually), when \(x\) was decreasing \(y\) was increasing. So, when \(x\) (in the reverse order) increases, \(y\) decreases. But the statement in B is wrong because adjacent \(y\) - values do not always differ by the same amount. For example, from \(y = 3\) to \(y = 6\) (difference \(=3\)), from \(y = 6\) to \(y = 7\) (difference \(=1\)).

Step3: Analyze option C

Since \(x\) and \(y\) have a trend (as \(x\) decreases \(y\) increases), the data shows a negative association, not no association.

Step4: Analyze option D

The \(x\) - value \(17\) occurs twice (\(x = 17\) corresponds to \(y = 7\) and \(y = 8\)). But having a repeated \(x\) - value does not mean no association. The overall trend (negative association) still exists.

Step5: Analyze option E

In the table, \(y\) increases from left - to - right (\(3,6,7,8,10,11,12\)), but this is because \(x\) is decreasing from left - to - right. The fact that \(y\) increases from left - to - right is a result of the negative association (as \(x\) decreases \(y\) increases), not the reason for no association.

Answer:

A. In the table, \(x\) decreases from left to right. Data with \(x\) - values that decrease from left to right on a table show negative association.