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.) On average, how much does the temperature decrease every five minutes?
To find the correlation coefficient and the average rate of temperature decrease, we can perform a linear regression analysis on the given data.
1.) The correlation coefficient, also known as Pearson's correlation coefficient, measures the strength and direction of the linear relationship between two variables. In this case, the variables are time (independent variable) and temperature (dependent variable). The correlation coefficient ranges from -1 to 1.
- A correlation coefficient of 1 indicates a perfect positive correlation, meaning that as the independent variable increases, the dependent variable also increases at a constant rate.
- A correlation coefficient of -1 indicates a perfect negative correlation, meaning that as the independent variable increases, the dependent variable decreases at a constant rate.
- A correlation coefficient of 0 indicates no linear relationship between the variables.
To find the correlation coefficient, we can use statistical software or tools like Excel, Python, or R. After performing the analysis, if the correlation coefficient is close to -1 or 1, it indicates a strong linear relationship between time and temperature in this case.
2.) To find the average rate of temperature decrease every five minutes, we need to determine the slope of the linear regression line. The slope represents the change in temperature for each unit of change in time.
Using the linear regression equation: temperature = slope * time + intercept, we can determine the slope value. In this case, the slope represents the average rate of temperature decrease per minute.
Once we have the slope value, we can multiply it by 5 to get the average rate of temperature decrease every five minutes.