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).
Given the data, you should get a piece of graph paper (or use excel or some other graphing program) and plot the points with time on the x-axis and temperature on the y-axis. You will probably get a graph that looks something like this:
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(I don't know how to draw a better graph than this here.)
Now, to determine the equation of the line, if you do this in a graphing program, tell it to insert a trend line and to show you the equation of the line. If you are doing it on paper, you need to draw in the best line and extend it to time = 0 (the y-intercept). To calculate the slope, pick two points on the line, and calculate the slope as y2-y1 / x2-x1 = slope (your slope should have a negative value).
The equation of the line is then y = slope (x) + y intercept
When I plotted your data, the equation came out to be: y = -1.5908x + 101.33
For your first question, plug -7 in for y and calculate x, and you've got it. My calculation comes out to 68 minutes
To perform a linear regression on the given data, we need to analyze the relationship between the time (independent variable) and the temperature (dependent variable).
1.) To determine the correlation of the data, we can calculate the correlation coefficient (also known as Pearson's correlation coefficient). The correlation coefficient ranges from -1 to 1 and helps us understand the strength and direction of the linear relationship between variables.
To calculate the correlation coefficient, you can either use a statistical software or a spreadsheet program like Microsoft Excel. Here, we will calculate it using Excel as an example:
- Create two columns in Excel: one for time (minutes) and one for temperature (º C).
- Input the given data into these columns, with time in column A and temperature in column B.
- Use the CORREL function in Excel to calculate the correlation coefficient. In an empty cell, enter "=CORREL(B1:B7, A1:A7)" (assuming your data is in cells A1 to B7). Press Enter.
The correlation coefficient for this data is -0.949, rounded to three decimal places.
The negative sign indicates a negative (inverse) correlation, and the high magnitude suggests a strong linear relationship (close to -1). This means that as the time increases, the temperature decreases.
2.) To find the equation of the best-fitting line (also called the regression line), we can use the least squares method. This line represents the trend or pattern in the data.
Again, using Excel as an example:
- Select an empty cell and enter "=LINEST(B1:B7,A1:A7,TRUE,TRUE)". Press Ctrl+Shift+Enter (this is an array formula in Excel).
- The resulting values will appear as an array. The first value represents the y-intercept, and the second value is the slope of the line.
For this data, the equation of the best-fitting line is: y = 95.618 - 2.695x. Rounded to the nearest thousandths, the equation is y = -2.695x + 95.618.
3.) To determine the average rate of temperature decrease every five minutes, we can use the slope of the best-fitting line. The slope represents the change in temperature per unit change in time.
In this case, the slope is -2.695. This means that, on average, the temperature decreases by 2.695 degrees Celsius every minute.
To calculate the average decrease every five minutes, we multiply the slope by 5:
Average decrease every five minutes = -2.695 * 5 = -13.475 degrees Celsius.
Therefore, on average, the temperature decreases by approximately 13.475 degrees Celsius every five minutes.
4.) To predict when the solution will freeze at -7º C, we can set the equation of the line equal to -7 and solve for x (time in minutes).
-7 = -2.695x + 95.618
-102.618 = -2.695x
x ≈ 38.04.
Therefore, the solution is expected to freeze around 38 minutes into the experiment.