Natalie performs a chemistry experiment where she records the temperature of an ongoing reaction. The solution is 93.5º C after 3 minutes; 90º C after 5 minutes, 84.8 C after 9 minutes; 70.2º C after 18 minute; 54.4º C after 30 minutes; 42.5ºC after 37 minutes; and 24.9º C after 48 minutes. Perform a linear regression on this data to complete the following items.
1.) What does the value of the correlation coefficient tell you about correlation of the data?
2.) Write the equation of the best-fitting line. (Round to the nearest thousandths.)
3.) On average, how much does the temperature decrease every five minutes?
4.) If Natalie's solution is expected to freeze at -7º C, how many minutes into the experiment should the solution freeze? (Show work that supports your prediction).
To perform linear regression and answer the questions, we will use the given data points and their corresponding time values. Here are the steps to solve each item:
1) To find the correlation coefficient, we will calculate the correlation between the time and temperature values. The correlation coefficient, often denoted as "r", represents the strength and direction of the linear relationship between two variables.
2) To find the equation of the best-fitting line (in the form y = mx + b), we will use the values obtained from the linear regression in step 1. The slope (m) represents the rate of temperature change over time, and the y-intercept (b) represents the starting temperature.
3) To calculate the average temperature decrease every five minutes, we will use the slope (m) obtained from the equation of the best-fitting line. Since the slope represents the rate of decrease, we can divide it by 5 to get the average temperature decrease per 5 minutes.
4) To determine when the solution is expected to freeze at -7º C, we will use the equation of the best-fitting line. We need to find the time (x-value) when the temperature (y-value) reaches -7º C. By substituting -7º C as the temperature in the equation, we can solve for the corresponding time value.
Let's now solve each item step by step:
1) To find the correlation coefficient:
- Gather the data pairs (time, temperature): (3, 93.5), (5, 90), (9, 84.8), (18, 70.2), (30, 54.4), (37, 42.5), (48, 24.9).
- Calculate the correlation coefficient (r), either by hand or using tools like Microsoft Excel or Google Sheets.
2) To get the equation of the best-fitting line:
- Use the values obtained from the correlation calculation in step 1.
- Substitute the slope (m) and y-intercept (b) values into the equation y = mx + b, rounding to the nearest thousandths.
3) To find the average temperature decrease every five minutes:
- Use the slope (m) obtained from the equation of the best-fitting line.
- Divide the slope by 5 to obtain the average temperature decrease per 5 minutes, rounding to an appropriate precision.
4) To determine when the solution would freeze at -7º C:
- Substitute -7º C as the temperature (y-value) into the equation of the best-fitting line.
- Solve for the corresponding time (x-value).
- Show the work that supports your prediction by displaying the equation, substituting the values, and solving for the time.
By following these steps, you can find the answers to the given questions using linear regression on the provided data.
To perform a linear regression on the given data, we can use the method of least squares to find the equation of the best-fitting line. Let's calculate the values step by step.
1) Correlation Coefficient:
The correlation coefficient, also known as r, measures the strength and direction of the relationship between two variables. It ranges from -1 to 1. A value of 1 indicates a perfect positive correlation, -1 indicates a perfect negative correlation, and 0 indicates no correlation.
To calculate the correlation coefficient, we can use the following formula:
r = Σ((x - x̄)(y - ȳ)) / √(Σ(x - x̄)² * Σ(y - ȳ)²)
Where:
- x is the time (in minutes)
- y is the temperature (in ºC)
- x̄ is the mean of the time values
- ȳ is the mean of the temperature values
- Σ represents the sum of the values
Using the given data, we find:
x: 3, 5, 9, 18, 30, 37, 48
y: 93.5, 90, 84.8, 70.2, 54.4, 42.5, 24.9
Calculating the means:
x̄ = (3 + 5 + 9 + 18 + 30 + 37 + 48) / 7 = 24
ȳ = (93.5 + 90 + 84.8 + 70.2 + 54.4 + 42.5 + 24.9) / 7 ≈ 66.7
Now we can calculate the numerator and denominators of the formula:
Numerator:
Σ((x - x̄)(y - ȳ)) = (3 - 24)(93.5 - 66.7) + (5 - 24)(90 - 66.7) + (9 - 24)(84.8 - 66.7) + (18 - 24)(70.2 - 66.7) + (30 - 24)(54.4 - 66.7) + (37 - 24)(42.5 - 66.7) + (48 - 24)(24.9 - 66.7) ≈ -1445.4
Denominators:
Σ(x - x̄)² = (3 - 24)² + (5 - 24)² + (9 - 24)² + (18 - 24)² + (30 - 24)² + (37 - 24)² + (48 - 24)² = 1402
Σ(y - ȳ)² = (93.5 - 66.7)² + (90 - 66.7)² + (84.8 - 66.7)² + (70.2 - 66.7)² + (54.4 - 66.7)² + (42.5 - 66.7)² + (24.9 - 66.7)² ≈ 5583.1
Now we can calculate the correlation coefficient:
r = (-1445.4) / √(1402 * 5583.1) ≈ -0.935
The correlation coefficient of approximately -0.935 indicates a strong negative correlation between time and temperature.
2) Equation of the best-fitting line:
The equation of a line is given by y = mx + b, where m is the slope and b is the y-intercept. In this case, we need to find the values of m and b using the least squares method.
First, let's calculate the slope (m):
m = r * (Σ(x - x̄) / Σ(y - ȳ))
m = (-0.935) * (Σ(x - x̄) / Σ(y - ȳ))
m = (-0.935) * (1402 / 5583.1) ≈ -0.234
Next, let's calculate the y-intercept (b):
b = ȳ - m * x̄
b = 66.7 - (-0.234) * 24 ≈ 72.5
Therefore, the equation of the best-fitting line is:
y ≈ -0.234x + 72.5
3) Average temperature decrease every five minutes:
From the equation of the best-fitting line, we can see that the slope (-0.234) represents the change in temperature for every minute. To find the average temperature decrease every five minutes, we multiply the slope by 5:
Average temperature decrease every five minutes = -0.234 * 5 ≈ -1.17º C
On average, the temperature decreases by approximately 1.17º C every five minutes.
4) Prediction for freezing point:
To find out when the solution will freeze, we need to determine the time when the temperature reaches -7º C. We can use the equation of the best-fitting line and solve for x (time):
-7 = -0.234x + 72.5
-0.234x = -7 - 72.5
-0.234x = -79.5
x ≈ -79.5 / -0.234 ≈ 340
Therefore, the solution is expected to freeze at approximately 340 minutes into the experiment.