In a population of exam scores, a score of X � 48
corresponds to z��1.00 and a score of X � 36
corresponds to z � –0.50. Find the mean and standard
deviation for the population. (Hint: Sketch the
distribution and locate the two scores on your sketch.)
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.
Cannot sketch on the posts.
Use same processes as indicated on my response to your previous post.
To find the mean and standard deviation for the population, we can use the formula for z-score:
z = (X - μ) / σ
where X is the score, μ is the mean, and σ is the standard deviation.
Given that a score of X = 48 corresponds to z = 1.00 and a score of X = 36 corresponds to z = -0.50, we can set up two equations:
1.00 = (48 - μ) / σ
-0.50 = (36 - μ) / σ
We can solve these two equations simultaneously to find the values of μ and σ.
From the first equation, we have:
1.00σ = 48 - μ (equation 1)
From the second equation, we have:
-0.50σ = 36 - μ (equation 2)
To solve for μ, we can rearrange equation 2:
μ = 36 + 0.50σ
Substituting this value of μ into equation 1, we get:
1.00σ = 48 - (36 + 0.50σ)
Simplifying the equation, we have:
1.00σ + 0.50σ = 12
1.50σ = 12
σ = 12 / 1.50
σ = 8
Now, substituting this value of σ into equation 1, we can solve for μ:
1.00(8) = 48 - μ
8 = 48 - μ
μ = 48 - 8
μ = 40
Therefore, the mean (μ) for the population is 40 and the standard deviation (σ) is 8.
For the sample consisting of scores: 2, 7, 4, 6, 4, and 7, we can compute the mean and standard deviation using the following formulas:
Mean (μ) = (2 + 7 + 4 + 6 + 4 + 7) / 6 = 30 / 6 = 5
Subtract the mean from each score and square the result:
(2 - 5)^2 = 9
(7 - 5)^2 = 4
(4 - 5)^2 = 1
(6 - 5)^2 = 1
(4 - 5)^2 = 1
(7 - 5)^2 = 4
Next, calculate the sum of the squared differences:
9 + 4 + 1 + 1 + 1 + 4 = 20
Divide the sum by the sample size minus 1:
20 / (6 - 1) = 20 / 5 = 4
Finally, take the square root to find the standard deviation:
Standard deviation (σ) = √4 = 2
Therefore, the mean (μ) for the sample is 5 and the standard deviation (σ) is 2.
To find the z-score for each score in the sample, we can use the formula:
z = (X - μ) / σ
For the sample scores:
z1 = (2 - 5) / 2 = -1.5
z2 = (7 - 5) / 2 = 1
z3 = (4 - 5) / 2 = -0.5
z4 = (6 - 5) / 2 = 0.5
z5 = (4 - 5) / 2 = -0.5
z6 = (7 - 5) / 2 = 1
Therefore, the z-scores for the sample scores are: -1.5, 1, -0.5, 0.5, -0.5, and 1.
To transform the original sample into a new sample with a mean of M = 50 and s = 10, we can use the formula:
X' = M + (X - μ) * (s / σ)
For each score in the original sample:
X1' = 50 + (2 - 5) * (10 / 2) = 50 - (3 * 5) = 50 - 15 = 35
X2' = 50 + (7 - 5) * (10 / 2) = 50 + (2 * 5) = 50 + 10 = 60
X3' = 50 + (4 - 5) * (10 / 2) = 50 - (1 * 5) = 50 - 5 = 45
X4' = 50 + (6 - 5) * (10 / 2) = 50 + (1 * 5) = 50 + 5 = 55
X5' = 50 + (4 - 5) * (10 / 2) = 50 - (1 * 5) = 50 - 5 = 45
X6' = 50 + (7 - 5) * (10 / 2) = 50 + (2 * 5) = 50 + 10 = 60
Therefore, the new sample with a mean of M = 50 and s = 10 is: 35, 60, 45, 55, 45, and 60.
To find the mean and standard deviation for the population in question, we need to make use of the given information about the scores and their corresponding z-scores.
First, let's sketch the distribution:
X: |-----------------|------------------|-------------|
36 X 48
-0.50 0 1.00
From the sketch, we can see that the mean (X-bar) corresponds to a z-score of 0.
We can use the formula for z-score to find the standard deviation (s):
z = (X - X-bar) / s
Given:
X = 48, z = 1.00
X = 36, z = -0.50
We can set up two equations:
1.00 = (48 - X-bar) / s ---(1)
-0.50 = (36 - X-bar) / s ---(2)
Solving equations (1) and (2) simultaneously will give us the values of X-bar and s.
Now, let's move on to the second part of the question about the sample:
a. To compute the mean (X-bar) for the sample, we add up all the scores and divide by the number of scores:
X-bar = (2 + 7 + 4 + 6 + 4 + 7) / 6
b. To calculate the standard deviation (s) for the sample, we need to find the variance first. Variance is the average of squared deviations from the mean:
variance = ((2 - X-bar)^2 + (7 - X-bar)^2 + (4 - X-bar)^2 + (6 - X-bar)^2 + (4 - X-bar)^2 + (7 - X-bar)^2) / 6
s = sqrt(variance) (taking the square root of the variance)
c. To transform the original sample into a new sample with a mean of 50 (M) and standard deviation of 10 (s), we can use the z-score formula:
z = (X - M) / s
For each score in the original sample, calculate the z-score using this formula.
Then, multiply each z-score by the new standard deviation (10) and add the new mean (50) to obtain the transformed scores.