Question

3. The table below lists the sound intensities and the decibel scale values.

Sound Intensity	316	500	750	2000	5000
Scale Value	25	27	28.75	33	37

a. Construct a scatterplot and select the appropriate model(s).

b. Find R² for each selected model(s) and write down the equation for the model with the largest R².

c. Use the equation in (b) to predict the scale value for the sound intensity of 9000.

Answer 1

a)

Scatter plot between sound intensities and the decibel scale values

The scatter plot is obtained in excel by following these steps,

Step 1: Write the data values in excel. The screenshot is shown below,

Step 2: select the Sound Intensity and Scale Value column then INSERT > Recommended Charts > Scatter > OK. The screenshot is shown below,

The Chart is obtained. The screenshot is shown below,

Appropriate model

1. logarithmic regression model

Since we want to predict the scale value for the sound intensity based, the regression model is used. From the scatterplot we can see that there is a logarithmic curved pattern hence the logarithmic regression model is used.

The regression model is defined as follows,

where Y = scale value, X = sound intensity

2. Quadratic regression model

The quadratic regression model can also be used for the this curve pattern.

The regression model is defined as follows,

b)

Answer:

Logarithmic regression model

R square = 0.999997

The regression model is,

Quadratic regression model

The R square value = 0.987572

The regression model is,

Explanation:

Logarithmic regression model

The regression analysis is done in excel by following these steps,

Step 1: Write the data values in excel.

Step 2: The natural log for independent variable X is obtained using the function =ln(). The screenshot is shown below,

Step 3: DATA > Data Analysis > Regression > OK. The screenshot is shown below,

Step 3: Select Input Y Range: 'Y' column, Input X Range: 'ln(X)' column then OK. The screenshot is shown below,

The result is obtained. The screenshot is shown below,

From the regression output summary,

The R square value = 0.999997

The regression model is,

Quadratic regression model

The regression analysis is done in excel by following these steps,

Step 1: Write the data values in excel.

Step 2: The square for independent variable X is obtained in excel. The screenshot is shown below,

Step 3: DATA > Data Analysis > Regression > OK. The screenshot is shown below,

Step 4: Select Input Y Range: 'Y' column, Input X Range: 'X and X^2' column then OK. The screenshot is shown below,

The result is obtained. The screenshot is shown below,

From the regression output summary,

The R square value = 0.987572

The regression model is,

c)

Answer:

The predicted scale value = 39.5443

Explanation:

The R square value for logarithmic model is larger compared to the quadratic regression model hence the logarithmic model is used to predict the the scale value,

The regression model is,

For X = 9000