Question

the table shows the gpa of several students and the number of hours they spend watching television each week. based on the table, what is the best prediction of a students gpa if the student watches 14 hours of television each week? 2.1 2.3 2.7 3.3

Explanation:

Step1: Analyze the trend

As the number of hours of TV watching (\(x\)) increases, the GPA (\(y\)) generally decreases. For example, when \(x = 3,y=3.5\); when \(x = 5,y = 3.4\); when \(x=10,y = 3.0\); when \(x = 12,y=2.7\).

Step2: Estimate for \(x = 14\)

Since the trend is a decrease in GPA with an increase in TV - watching hours. When \(x = 12,y = 2.7\). If we assume a linear - like trend (even though with only 4 data points it's a rough estimate), as \(x\) increases from 12 to 14 (an increase of 2), and looking at previous differences (e.g., from \(x = 10\) to \(x = 12\), \(y\) decreased by \(3.0−2.7 = 0.3\)). A further increase in \(x\) would lead to a further decrease in \(y\). A value less than \(2.7\) is more likely. Among the options, \(2.1\) and \(2.3\) are less than \(2.7\). But if we consider the rate of change (not a strict calculation, just a trend - based guess), if we assume a somewhat consistent rate of decrease (for example, if we consider the change from \(x=10\) (\(y = 3.0\)) to \(x = 12\) (\(y=2.7\)): \(\Delta x=2,\Delta y=- 0.3\)). If we extend this (very roughly) for \(x = 14\) (another \(\Delta x = 2\) from \(x = 12\)), a decrease of about \(0.3\) more would be too much (since it's a non - linear real - world relationship), but a small decrease. If we consider that when \(x\) is larger, the impact might be more (but still a rough guess). If we look at the fact that when \(x\) increases, \(y\) decreases, and \(2.1\) is a more significant decrease than \(2.3\). But if we consider the data:
Let's calculate the mean of \(y\) values: \(\bar{y}=\frac{3.0 + 3.4+3.5 + 2.7}{4}=\frac{12.6}{4}=3.15\). The values of \(y\) are \(3.0,3.4,3.5,2.7\). As \(x\) increases, \(y\) decreases. When \(x = 12,y = 2.7\). A student who watches more TV (\(x = 14\)) will have a lower GPA than \(2.7\). Among the options \(2.1\) and \(2.3\), if we assume a slow rate of decrease (since the change from \(x = 10\) to \(x = 12\) is a decrease of \(0.3\) in \(y\) for an increase of \(2\) in \(x\)), a decrease of \(0.4\) (from \(y = 2.7\) to \(y=2.3\)) for an increase of \(2\) in \(x\) (from \(x = 12\) to \(x = 14\)) is a more reasonable rough estimate compared to a decrease of \(0.6\) (to \(y = 2.1\)).

Answer:

2.3