Working on "Inferences from Dependent Samples" trying to find the standard deviation of the difference data.
Differences: 24, 53, 100, 40, -10,71,77,70, 39, 50
n=10
d=51.4
sd=??? (Answer suppose to be 30.8336 but not coming close to this number at all)
To find the standard deviation of the differences, you can follow these steps:
Step 1: Calculate the mean difference (d-bar):
To find the mean difference, you need to sum up all the differences and divide it by the number of observations (n).
d-bar = (Σd) / n
= (24 + 53 + 100 + 40 - 10 + 71 + 77 + 70 + 39 + 50) / 10
= 424 / 10
= 42.4
Step 2: Calculate the squared differences for each observation:
For each observation, subtract the mean difference (d-bar) from the actual difference and square the result.
Squared differences = (d - d-bar)^2
Calculating squared differences for each observation:
(24 - 42.4)^2 = (-18.4)^2 = 338.56
(53 - 42.4)^2 = (10.6)^2 = 112.36
(100 - 42.4)^2 = (57.6)^2 = 3327.36
(40 - 42.4)^2 = (-2.4)^2 = 5.76
(-10 - 42.4)^2 = (-52.4)^2 = 2740.16
(71 - 42.4)^2 = (28.6)^2 = 820.96
(77 - 42.4)^2 = (34.6)^2 = 1197.16
(70 - 42.4)^2 = (27.6)^2 = 761.76
(39 - 42.4)^2 = (-3.4)^2 = 11.56
(50 - 42.4)^2 = (7.6)^2 = 57.76
Step 3: Calculate the sum of squared differences:
Add up all the squared differences calculated in Step 2.
Sum of squared differences = Σ[(d - d-bar)^2]
= 338.56 + 112.36 + 3327.36 + 5.76 + 2740.16 + 820.96 + 1197.16 + 761.76 + 11.56 + 57.76
= 9543.76
Step 4: Calculate the variance of the differences:
Divide the sum of squared differences by (n - 1), where n is the number of observations.
Variance = (Sum of squared differences) / (n - 1)
= 9543.76 / (10 - 1)
= 9543.76 / 9
= 1060.42
Step 5: Calculate the standard deviation of the differences:
Take the square root of the variance calculated in Step 4 to get the standard deviation.
Standard deviation = √(Variance)
= √1060.42
= 30.8336 (rounded to four decimal places)
Therefore, the standard deviation of the differences is approximately 30.8336.