To determine whether the data show a linear relationship, let's analyze the given data points:
[tex]\[
\begin{tabular}{|l|c|c|c|c|c|}
\hline
\text{Hours, } x & 3 & 7 & 9 & 17 & 20 \\
\hline
\text{Battery life (\%), } y & 86 & 61 & 50 & 26 & 0 \\
\hline
\end{tabular}
\][/tex]
We need to check if these points can be roughly described by a linear equation of the form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept. Upon performing a linear regression analysis, we obtain the following results:
- Slope (m): -4.584661354581673
- Intercept (b): 95.94820717131472
Thus, the equation of the line that best fits the given data is approximately:
[tex]\[ y = -4.6x + 96 \][/tex]
This suggests that the battery life decreases linearly as the number of hours increases.
Now, let’s further analyze the implications of this linear relationship:
1. Determine after how many hours the battery life is 0%:
We solve for [tex]\( x \)[/tex] when [tex]\( y = 0 \)[/tex]:
[tex]\[
0 = -4.6x + 96 \\
x = \frac{96}{4.6} \\
x \approx 20.93
\][/tex]
So, the battery will be completely depleted (0% battery life remaining) after approximately 20.93 hours.
2. Determine the battery life after 10 hours:
Substituting [tex]\( x = 10 \)[/tex] into the equation [tex]\( y = -4.6x + 96 \)[/tex]:
[tex]\[
y = -4.6(10) + 96 \\
y = -46 + 96 \\
y = 50
\][/tex]
Therefore, after 10 hours, the battery life will be approximately 50%.
From the results, it is clear that the data do indeed show a linear relationship, accurately represented by the equation [tex]\( y = -4.6x + 96 \)[/tex].
