No page times or marks allowed. Redetermine lessons 1, 2, and 3.
Active the following that determine the degree of linear correlation:
Answers:
See the radiations to answer Questions 2 and 3.
Answer (1)
Calculate the mean of x values: x ˉ ≈ 64.93 .
Calculate the mean of y values: y ˉ ≈ 219.99 .
Calculate the standard deviations: s x ≈ 65.11 and s y ≈ 369.43 .
Calculate the Pearson correlation coefficient: r ≈ 0.615 , indicating a moderate positive correlation. The final answer is 0.615 .
Explanation
Understanding the Problem We are given two sets of data, x and y, and asked to determine the degree of linear correlation between them. This involves calculating the Pearson correlation coefficient, denoted as r, which measures the strength and direction of a linear relationship between two variables.
Calculating the Means First, we need to calculate the mean of the x values ( x ˉ ) and the mean of the y values ( y ˉ ). The x values are 10.4, 165, 22.9, 26.6, 33.8, 42.8, and 153. The y values are 118, 125, 15.7, 192, 21.9, 23.3, and 1044.
Mean of x The mean of x values is calculated as: x ˉ = 7 10.4 + 165 + 22.9 + 26.6 + 33.8 + 42.8 + 153 = 7 454.5 ≈ 64.93
Mean of y The mean of y values is calculated as: y ˉ = 7 118 + 125 + 15.7 + 192 + 21.9 + 23.3 + 1044 = 7 1539.9 ≈ 219.99
Calculating Standard Deviations Next, we calculate the standard deviation of x values ( s x ) and the standard deviation of y values ( s y ).
Standard Deviation of x The standard deviation of x values is approximately s x ≈ 65.11 .
Standard Deviation of y The standard deviation of y values is approximately s y ≈ 369.43 .
Calculating Pearson Correlation Coefficient Now, we calculate the Pearson correlation coefficient (r) using the formula: r = ( n − 1 ) s x s y ∑ i = 1 n ( x i − x ˉ ) ( y i − y ˉ ) Using the calculated values, we find that r ≈ 0.615 .
Interpreting the Result The Pearson correlation coefficient, r, is approximately 0.615. This value indicates a moderate positive linear correlation between the x and y values. A value of 0 indicates no linear correlation, while values close to +1 or -1 indicate strong positive or negative correlations, respectively.
Final Answer Therefore, the degree of linear correlation between the given x and y values is a moderate positive correlation.
Examples
Understanding the correlation between two variables is crucial in many real-world scenarios. For example, in finance, you might want to understand the correlation between the price of oil and the stock prices of energy companies. A positive correlation would suggest that as oil prices increase, the stock prices of energy companies also tend to increase. Similarly, in healthcare, you might want to understand the correlation between a patient's blood pressure and their cholesterol levels. Identifying these correlations helps in making informed decisions and predictions.
