[DATA] Crickets make a chirping noise by sliding their wings rapidly over each other. Perhaps you have noticed that the number of chirps seems to increase with the temperature. The following data list the temperature (in degrees Fahrenheit) and the number of chirps per second for the striped ground cricket.


c. Calculate the linear correlation coefficient between temperature and chirps per second.

Verified step by step guidance

1. Step 1: Organize the data into two lists or columns: one for temperature values (x) and one for chirps per second values (y). Make sure to include all paired data points from the table.
2. Step 2: Calculate the mean (average) of the temperature values, denoted as \(\bar{x}\), and the mean of the chirps per second values, denoted as \(\bar{y}\). Use the formulas: \(\bar{x} = \frac{1}{n} \sum_{i=1}^n x_i\) \(\bar{y} = \frac{1}{n} \sum_{i=1}^n y_i\) where \(n\) is the number of data points.
3. Step 3: Compute the deviations from the mean for each data point: \((x_i - \bar{x})\) and \((y_i - \bar{y})\). Then calculate the products of these deviations for each pair: \((x_i - \bar{x})(y_i - \bar{y})\).
4. Step 4: Calculate the sums needed for the correlation coefficient formula: \(S_{xy} = \sum (x_i - \bar{x})(y_i - \bar{y})\), \(S_{xx} = \sum (x_i - \bar{x})^2\), \(S_{yy} = \sum (y_i - \bar{y})^2\).
5. Step 5: Use the formula for the linear correlation coefficient \(r\): \(r = \frac{S_{xy}}{\sqrt{S_{xx} \times S_{yy}}}\) This value measures the strength and direction of the linear relationship between temperature and chirps per second.

Key Concepts

Linear Correlation Coefficient

The linear correlation coefficient, often denoted as r, measures the strength and direction of a linear relationship between two quantitative variables. Its value ranges from -1 to 1, where values close to 1 or -1 indicate strong positive or negative linear relationships, respectively, and values near 0 indicate little to no linear correlation.

Paired Data and Variables

Paired data consist of two related measurements for each subject or observation, such as temperature and chirps per second in this case. Understanding that each pair represents linked values is essential for calculating correlation, as it assesses how changes in one variable correspond to changes in the other.

Calculation of Correlation Coefficient

Calculating the correlation coefficient involves using the formula that standardizes the covariance of the two variables by dividing it by the product of their standard deviations. This process requires computing means, deviations, and sums of squares for both variables to quantify their linear association accurately.