IDNLearn.com provides a user-friendly platform for finding answers to your questions. Join our community to receive timely and reliable responses to your questions from knowledgeable professionals.
Sagot :
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.
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.
Thank you for joining our conversation. Don't hesitate to return anytime to find answers to your questions. Let's continue sharing knowledge and experiences! IDNLearn.com has the solutions to your questions. Thanks for stopping by, and come back for more insightful information.
