Answered

Dayton collected data to show the relationship between the number of hours he practices and the number of errors he makes when playing a new piece of music. The table shows his data.

Practice Makes Better

\begin{tabular}{|l|c|c|c|c|c|c|c|c|}
\hline
\begin{tabular}{l}
Number \\
of hours
\end{tabular} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\
\hline
\begin{tabular}{l}
Number \\
of Errors
\end{tabular} & 36 & 34 & 30 & 31 & 23 & 16 & 11 & 5 \\
\hline
\end{tabular}

Which is the approximate slope of the line of best fit for the data?

A. [tex]$-5.5$[/tex]
B. [tex]$-4.5$[/tex]
C. [tex]$-2.0$[/tex]
D. [tex]$-1.0$[/tex]



Answer :

GinnyAnswer
To determine the approximate slope of the line of best fit for the data, we will perform a linear regression analysis. The data provided gives the number of hours of practice and the corresponding number of errors:

[tex]\[ \begin{array}{|c|c|c|c|c|c|c|c|c|} \hline \text{Number of hours} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline \text{Number of Errors} & 36 & 34 & 30 & 31 & 23 & 16 & 11 & 5 \\ \hline \end{array} \][/tex]

From the linear regression analysis, we obtain the following results:
- Slope ([tex]\( m \)[/tex]): [tex]\(-4.55\)[/tex] (approximately),
- Intercept ([tex]\( b \)[/tex]): [tex]\(43.71\)[/tex],
- Correlation coefficient ([tex]\( r \)[/tex]): [tex]\(-0.97\)[/tex],
- P-value: [tex]\(5.42 \times 10^{-5}\)[/tex],
- Standard error ([tex]\( \text{std\_err} \)[/tex]): [tex]\(0.45\)[/tex]

For the slope of the line of best fit, we focus on the slope value, which is approximately [tex]\(-4.55\)[/tex]. This provides the rate at which the number of errors decreases for each additional hour of practice.

Given the following options:
- [tex]$-5.5$[/tex]
- [tex]$-4.5$[/tex]
- [tex]$-2.0$[/tex]
- [tex]$-1.0$[/tex]

The closest slope value to our calculated slope, [tex]\(-4.55\)[/tex], is [tex]\(-4.5\)[/tex].

Thus, the approximate slope of the line of best fit for the data is [tex]\(-4.5\)[/tex].

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