A sample consists of the following n � 6 scores: 2, 7,
4, 6, 4, and 7.
a. Compute the mean and standard deviation for the
sample.
b. Find the z-score for each score in the sample.
c. Transform the original sample into a new sample
with a mean of M � 50 and s � 10.
a. To compute the mean, we sum up all the scores and then divide by the total number of scores (n).
Mean = (2 + 7 + 4 + 6 + 4 + 7) / 6
Mean = 30 / 6
Mean = 5
To compute the standard deviation, we need to find the deviation of each score from the mean, square it, sum up all the squared deviations, divide by n-1, and then take the square root of that result.
Deviation from mean for each score:
(2 - 5) = -3
(7 - 5) = 2
(4 - 5) = -1
(6 - 5) = 1
(4 - 5) = -1
(7 - 5) = 2
Sum of squared deviations = (-3)^2 + 2^2 + (-1)^2 + 1^2 + (-1)^2 + 2^2
Sum of squared deviations = 9 + 4 + 1 + 1 + 1 + 4
Sum of squared deviations = 20
Standard deviation = √(Sum of squared deviations / (n - 1))
Standard deviation = √(20 / (6 - 1))
Standard deviation = √(20 / 5)
Standard deviation = √4
Standard deviation = 2
b. To find the z-score for each score, we use the formula:
z = (x - mean) / standard deviation
For the given scores:
z1 = (2 - 5) / 2
z2 = (7 - 5) / 2
z3 = (4 - 5) / 2
z4 = (6 - 5) / 2
z5 = (4 - 5) / 2
z6 = (7 - 5) / 2
Simplifying:
z1 = -3/2
z2 = 1/2
z3 = -1/2
z4 = 1/2
z5 = -1/2
z6 = 1/2
c. To transform the original sample into a new sample with a mean of M = 50 and standard deviation s = 10, we use the formula:
new score = (old score - mean) * (new standard deviation / old standard deviation) + new mean
For each score in the original sample:
new score = (old score - 5) * (10 / 2) + 50
So, the new transformed sample is:
new sample = [(2 - 5) * (10 / 2) + 50, (7 - 5) * (10 / 2) + 50, (4 - 5) * (10 / 2) + 50, (6 - 5) * (10 / 2) + 50, (4 - 5) * (10 / 2) + 50, (7 - 5) * (10 / 2) + 50]
Simplifying:
new sample = [40, 60, 45, 55, 45, 60]
a. To find the mean of a sample, you add up all the scores in the sample and divide the sum by the number of scores:
Mean = (2 + 7 + 4 + 6 + 4 + 7) / 6
Mean = 30 / 6
Mean = 5
So, the mean of the sample is 5.
To find the standard deviation of a sample, you need to calculate the variance first. The variance is the average of the squared differences of each score from the mean.
First, calculate the squared difference for each score:
(2 - 5)^2 = 9
(7 - 5)^2 = 4
(4 - 5)^2 = 1
(6 - 5)^2 = 1
(4 - 5)^2 = 1
(7 - 5)^2 = 4
Then, find the sum of these squared differences:
9 + 4 + 1 + 1 + 1 + 4 = 20
Next, divide the sum by the number of scores minus 1 (in this case, 6 - 1 = 5) to get the variance:
Variance = 20 / 5
Variance = 4
Finally, take the square root of the variance to find the standard deviation:
Standard Deviation = √(4)
Standard Deviation = 2
So, the standard deviation of the sample is 2.
b. To find the z-score for each score in the sample, use the formula:
z = (x - mean) / standard deviation
For the first score (2):
z = (2 - 5) / 2
z = -3 / 2
z = -1.5
For the second score (7):
z = (7 - 5) / 2
z = 2 / 2
z = 1
For the third score (4):
z = (4 - 5) / 2
z = -1 / 2
z = -0.5
For the fourth score (6):
z = (6 - 5) / 2
z = 1 / 2
z = 0.5
For the fifth score (4):
z = (4 - 5) / 2
z = -1 / 2
z = -0.5
For the sixth score (7):
z = (7 - 5) / 2
z = 2 / 2
z = 1
So, the z-scores for the scores in the sample are -1.5, 1, -0.5, 0.5, -0.5, and 1.
c. To transform the original sample into a new sample with a mean of 50 and a standard deviation of 10, you use the formula:
new_score = (old_score - old_mean) * (new_sd / old_sd) + new_mean
Using this formula for each score in the original sample:
For the first score (2):
new_score = (2 - 5) * (10 / 2) + 50
new_score = -3 * 5 + 50
new_score = -15 + 50
new_score = 35
For the second score (7):
new_score = (7 - 5) * (10 / 2) + 50
new_score = 2 * 5 + 50
new_score = 10 + 50
new_score = 60
For the third score (4):
new_score = (4 - 5) * (10 / 2) + 50
new_score = -1 * 5 + 50
new_score = -5 + 50
new_score = 45
For the fourth score (6):
new_score = (6 - 5) * (10 / 2) + 50
new_score = 1 * 5 + 50
new_score = 5 + 50
new_score = 55
For the fifth score (4):
new_score = (4 - 5) * (10 / 2) + 50
new_score = -1 * 5 + 50
new_score = -5 + 50
new_score = 45
For the sixth score (7):
new_score = (7 - 5) * (10 / 2) + 50
new_score = 2 * 5 + 50
new_score = 10 + 50
new_score = 60
So, the transformed sample with a mean of 50 and a standard deviation of 10 is: 35, 60, 45, 55, 45, 60.
a. Find the mean first = sum of scores/number of scores
Subtract each of the scores from the mean and square each difference. Find the sum of these squares. Divide that by the number of scores to get variance.
Standard deviation = square root of variance
b. Z = (score-mean)/SD
c. You can make up your own.
I'll let you do the calculations.