To find the best prediction of the wavelength of the key that is 8 keys above the A above middle C, let's follow these steps:
1. Data Summary and Preparation: We have the following data points:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Number of keys}, x & \text{Wavelength} (\text{cm}), y \\
\hline
0 & 78.41 \\
\hline
2 & 69.85 \\
\hline
3 & 65.93 \\
\hline
6 & 55.44 \\
\hline
10 & 44.01 \\
\hline
\end{array}
\][/tex]
2. Logarithmic Transformation: When performing exponential regression, convert the dependent variable [tex]\( y \)[/tex] (wavelength) using the natural logarithm [tex]\( \ln(y) \)[/tex].
3. Linear Regression on the Transformed Data: Perform a linear regression on the data [tex]\((x, \ln(y))\)[/tex] to find the linear relationship:
[tex]\[
\text{ln}(y) = \text{slope} \cdot x + \text{intercept}
\][/tex]
From the regression analysis, we have:
- Slope: [tex]\( -0.05775194578546561 \)[/tex]
- Intercept: [tex]\( 4.3618808452540865 \)[/tex]
4. Make Prediction for [tex]\( x = 8 \)[/tex]:
Substitute [tex]\( x = 8 \)[/tex] into the linear regression equation to find the predicted [tex]\(\text{ln}(y)\)[/tex]:
[tex]\[
\text{ln}(y)_{\text{pred}} = (-0.05775194578546561) \times 8 + 4.3618808452540865
\][/tex]
This simplifies to:
[tex]\[
\text{ln}(y)_{\text{pred}} = 3.8998652789703616
\][/tex]
5. Convert Back to Original Scale: Convert the [tex]\(\text{ln}(y)\)[/tex] value back to [tex]\( y \)[/tex] by exponentiating it:
[tex]\[
y_{\text{pred}} = e^{3.8998652789703616}
\][/tex]
This yields the predicted wavelength:
[tex]\[
y_{\text{pred}} = 49.39579400502108 \, \text{cm}
\][/tex]
6. Choose the Closest Answer:
Looking at the possible choices:
- 49.31 cm
- 49.44 cm
- 49.73 cm
- 49.78 cm
The best prediction of the wavelength of the key that is 8 keys above the A above middle C is:
[tex]\[
\boxed{49.44 \, \text{cm}}
\][/tex]
This value is the closest to the calculated predicted wavelength of approximately 49.39579400502108 cm.
