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Answer :
The scatter diagram accurately represents the data set by showing the relationship between x and y. By analyzing the scatter plot, we can determine the strength, direction, and nature of the correlation. Additionally, by computing the sample covariance and correlation coefficient, we can quantitatively measure the relationship between the variables.
a. To determine which scatter diagram accurately represents the data set, we need to analyze each option and consider the relationship between the variables x and y. A scatter diagram is a graphical representation of the relationship between two variables. It consists of individual data points plotted on the graph, which helps us understand the correlation between the variables. By examining the shape of the scatter plot and the pattern of the data points, we can identify the strength and direction of the relationship between x and y.
b. The scatter diagram indicates the relationship between x and y. If the data points on the scatter plot show a general upward trend, it suggests a positive relationship between the variables. Conversely, if the data points display a downward trend, it indicates a negative relationship. If there is no clear pattern or trend, it suggests no relationship or a weak relationship between x and y.
c. To compute the sample covariance, we need the individual data points of x and y. The formula for sample covariance is:
cov(x,y) = Σ((xi - x')(yi - y'))/(n-1), where xi and yi are the individual data points, x' and y' are the means of x and y respectively, and n is the sample size.
d. To compute the sample correlation coefficient, we can use the formula:
r = cov(x,y) / (sx * sy), where cov(x,y) is the sample covariance, sx is the sample standard deviation of x, and sy is the sample standard deviation of y. The correlation coefficient, r, ranges from -1 to 1. A positive value indicates a positive correlation, a negative value indicates a negative correlation, and a value close to 0 suggests no correlation.
Based on the computation of the sample correlation coefficient, we can conclude the strength and direction of the relationship between x and y. If the correlation coefficient is close to 1 or -1, it indicates a strong relationship. If the coefficient is close to 0, it suggests a weak or no relationship between the variables.
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