A sample of 148 of our statistics students rated their level of admiration for Hillary Rodham Clinton on a scale of 1 to 7. The mean rating was 4.06, and the standard deviation was 1.70. (For this exercise, treat this sample as the entire population of interest.)
a. Use these data to demonstrate that the mean of thez distribution is always 0.
b. Use these data to demonstrate that the standard deviation of the z distribution is always 1.
c. Calculate the z score for a student who rated his admiration of Hillary Rodham Clinton as 6.1.
d. A student had a z score of �0.55. What rating did she give for her admiration of Hillary Rodham Clinton?
To answer these questions, we need to understand what a z-score is and how it is calculated.
A z-score, also known as a standard score, measures how many standard deviations an individual data point is away from the mean of a distribution. It helps us understand the relative position of a data point in the distribution.
a. To demonstrate that the mean of the z-distribution is always 0, we need to calculate the z-score for each data point in the given sample and then calculate the mean of those z-scores.
To calculate the z-score for each data point, we use the formula:
z = (x - μ) / σ
Where:
z = z-score
x = individual data point
μ = mean of the distribution
σ = standard deviation of the distribution
Since the sample is treated as the entire population for this exercise, the mean of the distribution μ is equal to the sample mean (4.06) and the standard deviation σ is equal to the sample standard deviation (1.70).
Now, for each data point, calculate the z-score using the formula above. Subtract the mean of the distribution from each data point and then divide by the standard deviation:
For the first data point (x = 1):
z1 = (1 - 4.06) / 1.70
z1 = -1.8
For the second data point (x = 2):
z2 = (2 - 4.06) / 1.70
z2 = -1.20
Continue this process for all 148 data points.
Next, calculate the mean of these z-scores.
Mean of z-scores = (z1 + z2 + ... + z148) / 148
Since we are treating the sample as the entire population, the mean of the z-distribution will always be 0. If you calculate the mean of the z-scores, you will find that it is indeed 0.
b. Similarly, to demonstrate that the standard deviation of the z-distribution is always 1, we need to calculate the standard deviation of the calculated z-scores. The formula to calculate the standard deviation of a sample is:
σz = √(Σ(z - μz)^2 / N)
Where:
σz = standard deviation of the z-distribution
z = each calculated z-score
μz = mean of the z-scores (which is always 0)
N = number of data points (148 in this case)
Plug in the values and calculate σz. The result will be 1, demonstrating that the standard deviation of the z-distribution is always 1.
c. To calculate the z-score for a student who rated his admiration of Hillary Rodham Clinton as 6.1, we use the same formula as mentioned earlier:
z = (x - μ) / σ
Substitute the values:
z = (6.1 - 4.06) / 1.70
z = 1.2
Therefore, the z-score for the student who rated his admiration as 6.1 is 1.2.
d. To calculate the rating a student gave for her admiration of Hillary Rodham Clinton, given a z score of -0.55, we rearrange the formula:
z = (x - μ) / σ
Now, we solve for x:
x = (z * σ) + μ
Substitute the values:
x = (-0.55 * 1.70) + 4.06
x = 3.717
Therefore, the student rated her admiration of Hillary Rodham Clinton as approximately 3.717.
Keep in mind that these calculations assume a normal distribution and the sample is treated as the entire population.