Sure, let's go through this step-by-step to understand the problem and find the solution.
### Step 1: Entering the Values
First, you enter the given data into the table. Here are the corresponding values for hours slept ([tex]$x$[/tex]) and test scores ([tex]$y$[/tex]):
[tex]\[
\begin{array}{|c|c|c|c|c|c|}
\hline
x & 4 & 6 & 6 & 8 & 10 \\
\hline
y & 62 & 75 & 88 & 79 & 97 \\
\hline
\end{array}
\][/tex]
### Step 2: Plot Points on the Graph
Plot these (x, y) pairs on a two-dimensional graph:
- (4, 62)
- (6, 75)
- (6, 88)
- (8, 79)
- (10, 97)
### Step 3: Calculating the Trend Line
The trend line is found by fitting a linear regression model to the data points. The line of best fit (or linear regression line) is given by the equation:
[tex]\[ y = mx + b \][/tex]
where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept of the line.
From the given answer:
- Slope ([tex]\( m \)[/tex]): 4.865384615384619
- Y-intercept ([tex]\( b \)[/tex]): 47.115384615384606
### Step 4: Predicting the Score for 5 Hours of Sleep
The equation of the regression line we found is:
[tex]\[ y = 4.865384615384619x + 47.115384615384606 \][/tex]
To predict the score for 5 hours of sleep ([tex]\( x = 5 \)[/tex]):
[tex]\[ y = 4.865384615384619 \cdot 5 + 47.115384615384606 \][/tex]
Plug in the value of [tex]\( x \)[/tex]:
[tex]\[ y \approx (4.865384615384619 \times 5) + 47.115384615384606 \][/tex]
[tex]\[ y \approx 24.326923076923093 + 47.115384615384606 \][/tex]
[tex]\[ y \approx 71.44230769230771 \][/tex]
So, the expected test score after 5 hours of sleep is approximately 71.44.
### Step 5: Determining the Trend
To determine how the test scores change with the number of hours slept, observe the slope of the trend line. The positive slope (4.865384615384619) indicates that as the number of hours slept increases, the test score also increases.
Thus, as the number of hours sleeping increases, the corresponding test score increases.
### Summary
The answers to the questions are:
1. The expected test score after 5 hours of sleep is approximately 71.44.
2. As the number of hours sleeping increases, the corresponding test score increases.
