These data were generated using the equation y = 2x + 1. A scatter diagram of the data results in five points that fall perfectly on a straight line.

x 0 1 2 3 4
y 1 3 5 7 9

(a) Find the correlation coefficient.

1.0

When data fit exactly onto a straight line, the correlation coefficient is either +1 or -1.
If the slope of the line is positive, r=+1. If the slope of the line is negative, r=-1.
In this case, the slope=+2, so r=+1.0

To find the correlation coefficient, you need to calculate the covariance and the standard deviations of both the x and y variables. Let's break it down step by step:

Step 1: Calculate the mean (average) of the x values and the y values.
- For x: mean(x) = (0 + 1 + 2 + 3 + 4) / 5 = 2
- For y: mean(y) = (1 + 3 + 5 + 7 + 9) / 5 = 5

Step 2: Subtract the mean from each x value and y value to get the deviations from the mean.
- For x: deviations from mean(x) = (0 - 2), (1 - 2), (2 - 2), (3 - 2), (4 - 2)
= -2, -1, 0, 1, 2
- For y: deviations from mean(y) = (1 - 5), (3 - 5), (5 - 5), (7 - 5), (9 - 5)
= -4, -2, 0, 2, 4

Step 3: Calculate the product of the deviations for each pair of x and y values and sum them.
- Product of deviations: (-2 * -4) + (-1 * -2) + (0 * 0) + (1 * 2) + (2 * 4)
= 8 + 2 + 0 + 2 + 8 = 20

Step 4: Calculate the sum of the squared deviations of x and y values.
- For x: sum of squared deviations = (-2^2) + (-1^2) + (0^2) + (1^2) + (2^2)
= 4 + 1 + 0 + 1 + 4 = 10
- For y: sum of squared deviations = (-4^2) + (-2^2) + (0^2) + (2^2) + (4^2)
= 16 + 4 + 0 + 4 + 16 = 40

Step 5: Calculate the standard deviation for x and y.
- For x: standard deviation of x = sqrt(sum of squared deviations / (number of values - 1))
= sqrt(10 / 4) = sqrt(2.5)
- For y: standard deviation of y = sqrt(sum of squared deviations / (number of values - 1))
= sqrt(40 / 4) = sqrt(10)

Step 6: Calculate the covariance.
- Covariance = sum of product of deviations / (number of values - 1)
- Covariance = 20 / 4 = 5

Step 7: Calculate the correlation coefficient.
- Correlation coefficient = covariance / (standard deviation of x * standard deviation of y)
- Correlation coefficient = 5 / (sqrt(2.5) * sqrt(10))
- Correlation coefficient โ‰ˆ 1

Therefore, the correlation coefficient for these data points is approximately 1. The value of 1 indicates a perfect positive linear relationship between x and y, which aligns with the given information that the points fall perfectly on a straight line.