Consider the following set of data.
(18, 12), (29, 48), (65, 29), (82, 24), (115, 56), (124, 13)
(a) Calculate the covariance of the set of data. (Give your answer correct to two decimal places.)
Incorrect: Your answer is incorrect. .
(b) Calculate the standard deviation of the six x-values and the standard deviation of the six y-values. (Give your answers correct to three decimal places.)
sx =
sy =
(c) Calculate r, the coefficient of linear correlation, for the data in part (a). (Give your answer correct to two decimal places.)
a.
dbar = 433/6
var(x) = x^2/n -xbar^2
var(x) = 40715/6 -(433/6)^2 = 1577.81
ybar = 182/6
var(y) = y^2 -ybar^2
var(y) = 7170/6 -(182/6)^2 = 274.89
cov(xy) = (xy)/n -(xbar)(ybar)
cov(xy) = 13513/6 -(433/6)(182/6) = 63.11
b.
sx = 43.51
sy = 18.16
C. r = cov(xy/sqrt(var(x)sqrt(var(y))
r = 63.11/sqrt(1577.81)sqrt(274.89) = 0.0958
r = 0.10
To calculate the covariance of the set of data, follow these steps:
Step 1: Calculate the mean of the x-values.
To find the mean of x-values, add up all the x-values and divide the sum by the number of values. For this data set, the x-values are 18, 29, 65, 82, 115, and 124.
Mean of x-values = (18 + 29 + 65 + 82 + 115 + 124) / 6
Step 2: Calculate the mean of the y-values.
To find the mean of y-values, add up all the y-values and divide the sum by the number of values. For this data set, the y-values are 12, 48, 29, 24, 56, and 13.
Mean of y-values = (12 + 48 + 29 + 24 + 56 + 13) / 6
Step 3: Calculate the covariance.
The covariance formula is given by:
Covariance (x, y) = ∑((x - mean of x) * (y - mean of y)) / (number of values)
Using the given formula, you need to calculate the summation of all values of (x - mean of x) * (y - mean of y) and divide it by the number of values.
Once you have calculated the covariance, you can move on to calculating the standard deviation of x-values and y-values.
To calculate the standard deviation of a set of values, follow these steps:
Step 1: Calculate the variance.
The variance formula is given by:
Variance = ∑((x - mean)^2) / (number of values)
Using the variance formula, calculate the summation of all values of (x - mean)^2 and divide it by the number of values to find the variance.
Step 2: Calculate the standard deviation.
The standard deviation is the square root of the variance.
Standard Deviation = √(Variance)
Apply the above steps to the x-values and y-values to find the standard deviations of both.
Finally, to calculate the coefficient of linear correlation (r) for the data, use the formula:
Coefficient of Linear Correlation (r) = Covariance / (Standard Deviation of x * Standard Deviation of y)
After calculating r, round your answer to two decimal places.
Now you should have the complete solution for parts (a), (b), and (c) of the question.