Some college students collected data on the intensity of light at various depths in a lake. Here are their data:
Depth (meters) Light intenisty (lumens)
5 168.00
6 120.42
7 86.31
8 61.87
9 44.34
10 31.78
11 22.78
Take the natural logarithm (ln) of the light intensity measurements and calculate the least-squares regession equation for the transformed data to perform the inverse transformation to express light intensity as an exponential function of depth in the lake.
Well, it seems like these college students took their deep dive into data collection quite literally! Anyway, let's get down to business.
To find the least-squares regression equation for the transformed data, we'll start by taking the natural logarithm (ln) of the light intensity measurements. Here are the transformed data:
Depth (meters) Transformed Light Intensity (ln)
5 5.123
6 4.795
7 4.457
8 4.122
9 3.794
10 3.461
11 3.126
Now, let's calculate the regression equation for the transformed data. We'll use the formula:
ln(y) = a + b*x
where ln(y) is the natural logarithm of the light intensity, x is the depth, a is the y-intercept, and b is the slope of the regression line.
Using this formula, we can calculate the values of a and b. I'll do the math and give you the final equation. Give me a moment, please.
*Calculating, calculating... almost there...*
Alright, the least-squares regression equation for the transformed data is:
ln(y) = 0.618 - 0.169*x
Now, to perform the inverse transformation and express light intensity as an exponential function of depth in the lake, we'll "undo" the natural logarithm (ln).
The inverse transformation equation is:
y = e^(0.618 - 0.169*x)
where y is the light intensity, x is the depth, and e is the mathematical constant approximately equal to 2.71828.
So, to express light intensity as an exponential function of depth in the lake, you can use the equation:
y = e^(0.618 - 0.169*x)
Hope this helps you shed some light on your question! Keep swimming in the sea of knowledge!
To calculate the least-squares regression equation for the transformed data, we need to perform the following steps:
Step 1: Take the natural logarithm (ln) of the light intensity measurements.
The transformed data will look like:
Depth (meters) Transformed Light Intensity (ln)
5 5.123
6 4.791
7 4.455
8 4.123
9 3.792
10 3.461
11 3.122
Step 2: Calculate the regression equation for the transformed data.
Let's define our variables:
x = Depth (meters)
y = Transformed Light Intensity (ln)
We need to find the equation of the line that best fits the transformed data. This equation will have the form:
y = mx + b
where m is the slope and b is the y-intercept of the line.
Using the least-squares regression method, we can calculate the values of m and b using the following formulas:
m = (nΣ(xy) - ΣxΣy) / (nΣ(x^2) - (Σx)^2)
b = (Σy - mΣx) / n
where n is the number of data points.
Calculating the above values using the provided data:
n = 7 (number of data points)
Σx = 5 + 6 + 7 + 8 + 9 + 10 + 11 = 56
Σy = 5.123 + 4.791 + 4.455 + 4.123 + 3.792 + 3.461 + 3.122 = 28.89
Σxy = (5 × 5.123) + (6 × 4.791) + (7 × 4.455) + (8 × 4.123) + (9 × 3.792) + (10 × 3.461) + (11 × 3.122) = 165.703
Σ(x^2) = (5^2) + (6^2) + (7^2) + (8^2) + (9^2) + (10^2) + (11^2) = 511
Now we can calculate m:
m = (7 × 165.703 - 56 × 28.89) / (7 × 511 - 56^2) = -0.572
And we can calculate b:
b = (28.89 - (-0.572 × 56)) / 7 = 6.223
Step 3: Perform the inverse transformation to express light intensity as an exponential function of depth.
The least-squares regression equation for the transformed data is:
ln(y) = -0.572x + 6.223
To express light intensity (y) as an exponential function of depth (x), we can exponentiate both sides of the equation:
e^(ln(y)) = e^(-0.572x + 6.223)
y = e^(-0.572x) * e^(6.223)
The final exponential equation representing light intensity as a function of depth in the lake is:
y = e^(-0.572x) * 499.63
Note: The constant term e^(6.223) is approximately 499.63.
So, the exponential function representing light intensity as a function of depth in the lake is given by:
y = 499.63 * e^(-0.572x)
To calculate the least-squares regression equation for the transformed data, we will follow these steps:
Step 1: Calculate the natural logarithm (ln) of the light intensity measurements.
The natural logarithm (ln) of a number can be calculated using a calculator or software that has a logarithm function. Here are the ln values for each light intensity measurement:
Depth (meters) Light intensity (lumens) ln(Light intensity)
5 168.00 5.123
6 120.42 4.791
7 86.31 4.458
8 61.87 4.123
9 44.34 3.795
10 31.78 3.459
11 22.78 3.128
Step 2: Calculate the mean of the ln(light intensity) values.
Add up all the ln(light intensity) values and divide by the total number of data points. In this case, there are 7 data points, so:
Mean ln(Light intensity) = (5.123 + 4.791 + 4.458 + 4.123 + 3.795 + 3.459 + 3.128) / 7 = 4.263
Step 3: Calculate the differences between each ln(light intensity) value and the mean ln(Light intensity).
Subtract the mean ln(Light intensity) from each ln(Light intensity) value to get the differences:
Difference = ln(Light intensity) - Mean ln(Light intensity)
Depth (meters) Light intensity (lumens) ln(Light intensity) Difference
5 168.00 5.123 0.860
6 120.42 4.791 0.528
7 86.31 4.458 0.195
8 61.87 4.123 -0.140
9 44.34 3.795 -0.468
10 31.78 3.459 -0.804
11 22.78 3.128 -1.135
Step 4: Calculate the squared differences.
Square each difference value to get the squared differences:
Squared Difference = Difference^2
Depth (meters) Light intensity (lumens) ln(Light intensity) Difference Squared Difference
5 168.00 5.123 0.860 0.739
6 120.42 4.791 0.528 0.279
7 86.31 4.458 0.195 0.038
8 61.87 4.123 -0.140 0.020
9 44.34 3.795 -0.468 0.219
10 31.78 3.459 -0.804 0.646
11 22.78 3.128 -1.135 1.287
Step 5: Sum up the squared differences.
Add up all the squared differences to get the sum of squared differences:
Sum of squared differences = 0.739 + 0.279 + 0.038 + 0.020 + 0.219 + 0.646 + 1.287 = 3.228
Step 6: Calculate the variance.
Divide the sum of squared differences by (n-1), where n is the number of data points. In this case, n = 7:
Variance = Sum of squared differences / (n - 1) = 3.228 / (7 - 1) = 0.537
Step 7: Calculate the covariance.
The covariance is the product of the differences between the depth and the mean of depth (D - Mean Depth) and the differences between the ln(light intensity) and the mean of ln(light intensity) (ln(Light intensity) - Mean ln(Light intensity)). Here are the calculations:
Depth (meters) Light intensity (lumens) ln(Light intensity) Difference Squared Difference Depth - Mean Depth Covariance
5 168.00 5.123 0.860 0.739 -3 -2.517
6 120.42 4.791 0.528 0.279 -2 -0.558
7 86.31 4.458 0.195 0.038 -1 0.038
8 61.87 4.123 -0.140 0.020 0 0
9 44.34 3.795 -0.468 0.219 1 0.219
10 31.78 3.459 -0.804 0.646 2 1.292
11 22.78 3.128 -1.135 1.287 3 3.860
Covariance = (-2.517 * -3) + (-0.558 * -2) + (0.038 * -1) + (0 * 0) + (0.219 * 1) + (1.292 * 2) + (3.860 * 3) = 22.621
Step 8: Calculate the slope (b) of the regression equation.
The slope (b) of the regression equation can be calculated using the following formula:
b = covariance / variance
b = 22.621 / 0.537 = 42.132
Step 9: Calculate the intercept (a) of the regression equation.
The intercept (a) of the regression equation can be calculated using the following formula:
a = mean ln(Light intensity) - (b * mean depth)
a = 4.263 - (42.132 * 8) = - 339.569
Step 10: Write the regression equation.
The least-squares regression equation for the transformed data is:
ln(Light intensity) = -339.569 + 42.132 * Depth
To perform the inverse transformation and express light intensity as an exponential function of depth in the lake, we need to take the exponent of both sides of the equation:
Light intensity = e^(-339.569 + 42.132 * Depth)
where e is the base of the natural logarithm, approximately equal to 2.71828.
This equation represents how light intensity depends on the depth in the lake, based on the collected data.