Questions: How sensitive to changes in water temperature are coral reefs? To find out, scientists examined data on sea surface temperatures and coral growth per year at locations in the Gulf of Mexico and the Caribbean Sea. The table shows the data for the Gulf of Mexico. Sea surface temperature: 26.7, 26.6, 26.6, 26.5, 26.3, 26.1 Growth: 0.85, 0.85, 0.79, 0.86, 0.89, 0.92 Click to download the data in your preferred format. CSV Excel JMP Mac-Text Minitab14-18 Minitab18+ PC-Text R SPSS TT Crunchlt! Find the correlation r step by step. Let x denote the sea surface temperature and y denote the coral growth. First, find the mean and standard deviation of each variable. Give your answer to at least two decimal places but if you are calculating the values by hand, avoid rounding during intermediate steps, mean for the x data set, x̄ = degrees

Solution Steps

First, we calculate the mean of the sea surface temperature (x) and the coral growth (y).

For sea surface temperature (x): \[ \bar{x} = \frac{26.7 + 26.6 + 26.6 + 26.5 + 26.3 + 26.1}{6} = \frac{158.8}{6} = 26.47 \]

For coral growth (y): \[ \bar{y} = \frac{0.85 + 0.85 + 0.79 + 0.86 + 0.89 + 0.92}{6} = \frac{5.16}{6} = 0.86 \]

Next, we calculate the standard deviation of x and y.

For sea surface temperature (x): \[ s_x = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}} \] \[ s_x = \sqrt{\frac{(26.7-26.47)^2 + (26.6-26.47)^2 + (26.6-26.47)^2 + (26.5-26.47)^2 + (26.3-26.47)^2 + (26.1-26.47)^2}{5}} \] \[ s_x = \sqrt{\frac{0.0529 + 0.0169 + 0.0169 + 0.0009 + 0.0289 + 0.1369}{5}} \] \[ s_x = \sqrt{\frac{0.2534}{5}} = \sqrt{0.05068} \approx 0.23 \]

For coral growth (y): \[ s_y = \sqrt{\frac{\sum (y_i - \bar{y})^2}{n-1}} \] \[ s_y = \sqrt{\frac{(0.85-0.86)^2 + (0.85-0.86)^2 + (0.79-0.86)^2 + (0.86-0.86)^2 + (0.89-0.86)^2 + (0.92-0.86)^2}{5}} \] \[ s_y = \sqrt{\frac{0.0001 + 0.0001 + 0.0049 + 0 + 0.0009 + 0.0036}{5}} \] \[ s_y = \sqrt{\frac{0.0096}{5}} = \sqrt{0.00192} \approx 0.04 \]

Finally, we calculate the correlation coefficient (r) using the formula: \[ r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{(n-1) s_x s_y} \]

\[ r = \frac{(26.7-26.47)(0.85-0.86) + (26.6-26.47)(0.85-0.86) + (26.6-26.47)(0.79-0.86) + (26.5-26.47)(0.86-0.86) + (26.3-26.47)(0.89-0.86) + (26.1-26.47)(0.92-0.86)}{5 \times 0.23 \times 0.04} \]

\[ r = \frac{(-0.23 \times -0.01) + (0.13 \times -0.01) + (0.13 \times -0.07) + (0.03 \times 0) + (-0.17 \times 0.03) + (-0.37 \times 0.06)}{5 \times 0.23 \times 0.04} \]

\[ r = \frac{0.0023 + 0.0013 - 0.0091 + 0 - 0.0051 - 0.0222}{0.046} \]

\[ r = \frac{-0.0328}{0.046} \approx -0.71 \]

Final Answer