What is the correlation coefficient, r , for this data? x & y 34 & 59 57 & 53 62 & 55 73 & 58 96 & 32 97 & 48 Round your answer to the nearest tenth. [ r= ] square

# Explanation<br /><br />To find the correlation coefficient, \( r \), for the given data, we will use the Pearson correlation coefficient formula. The formula for \( r \) is given by:<br /><br />\[<br />r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}}<br />\]<br /><br />where:<br />- \( n \) is the number of data points,<br />- \( \sum xy \) is the sum of the product of each pair of \( x \) and \( y \) values,<br />- \( \sum x \) and \( \sum y \) are the sums of \( x \) and \( y \) values respectively,<br />- \( \sum x^2 \) and \( \sum y^2 \) are the sums of the squares of \( x \) and \( y \) values respectively.<br /><br />Given data:<br />\[<br />\begin{array}{|c|c|}<br\ />\hline\ x\ \ &\ \ y\ \ \\<br\ />\hline\ 34\ &\ 59\ \\<br\ />\hline\ 57\ &\ 53\ \\<br\ />\hline\ 62\ &\ 55\ \\<br\ />\hline\ 73\ &\ 58\ \\<br\ />\hline\ 96\ &\ 32\ \\<br\ />\hline\ 97\ &\ 48\ \\<br\ />\hline<br\ />\end{array}<br />\]<br /><br />First, we calculate the necessary sums:<br />- \( n = 6 \)<br />- \( \sum x = 34 + 57 + 62 + 73 + 96 + 97 = 419 \)<br />- \( \sum y = 59 + 53 + 55 + 58 + 32 + 48 = 305 \)<br />- \( \sum xy = (34 \times 59) + (57 \times 53) + (62 \times 55) + (73 \times 58) + (96 \times 32) + (97 \times 48) = 2006 + 3021 + 3410 + 4234 + 3072 + 4656 = 20399 \)<br />- \( \sum x^2 = 34^2 + 57^2 + 62^2 + 73^2 + 96^2 + 97^2 = 1156 + 3249 + 3844 + 5329 + 9216 + 9409 = 33203 \)<br />- \( \sum y^2 = 59^2 + 53^2 + 55^2 + 58^2 + 32^2 + 48^2 = 3481 + 2809 + 3025 + 3364 + 1024 + 2304 = 16007 \)<br /><br />Now, substitute these values into the formula:<br /><br />\[<br />r = \frac{6(20399) - (419)(305)}{\sqrt{[6(33203) - (419)^2][6(16007) - (305)^2]}}<br />\]<br /><br />\[<br />r = \frac{122394 - 127775}{\sqrt{[199218 - 175561][96042 - 93025]}}<br />\]<br /><br />\[<br />r = \frac{-5381}{\sqrt{23657 \times 3017}}<br />\]<br /><br />\[<br />r = \frac{-5381}{\sqrt{71391909}}<br />\]<br /><br />\[<br />r = \frac{-5381}{8449.4}<br />\]<br /><br />\[<br />r \approx -0.636<br />\]<br /><br />Rounding to the nearest tenth, \( r \approx -0.6 \).<br /><br /># Answer<br /><br />\[<br />r = -0.6<br />\]