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?

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a. To demonstrate that the mean of the z-distribution is always 0, we need to convert the sample data to z-scores.

The formula for calculating the z-score is:

z = (x - mean) / standard deviation

In this case, the mean rating is 4.06 and the standard deviation is 1.70.

Let's take the first rating, x = 1.

z = (1 - 4.06) / 1.70
z = -1.8

Similarly, we can calculate the z-scores for all 148 ratings in your sample. The sum of the z-scores will be 0, since for every rating below the mean, there will be a corresponding rating above the mean, canceling out the effects of each other.

b. To demonstrate that the standard deviation of the z-distribution is always 1, we need to calculate the sample standard deviation and convert it to a z-score.

The formula for calculating the sample standard deviation is:

standard deviation = √(sum((x - mean)^2) / (n - 1))

Using the provided data, we can calculate the sample standard deviation by substituting the values:

standard deviation = √(sum((x - 4.06)^2) / (148 - 1))

After calculating, the sample standard deviation is 1.70. Now, to convert it to a z-score, divide the standard deviation by itself:

z = standard deviation / standard deviation

z = 1.70 / 1.70 = 1

This shows that the standard deviation of the z-distribution is always 1.

c. To calculate the z-score for a student who rated their admiration of Hillary Rodham Clinton as 6.1, we use the z-score formula:

z = (x - mean) / standard deviation

Substituting the provided values:

z = (6.1 - 4.06) / 1.70
z = 1.20

So, a student who rated their admiration as 6.1 has a z-score of 1.20.

d. To find the rating given by a student who had a z-score of 0.55, we rearrange the z-score formula:

z = (x - mean) / standard deviation

Substituting the provided z-score of 0.55 and the mean of 4.06, we have:

0.55 = (x - 4.06) / 1.70

Now, solve for x:

x - 4.06 = 0.55 * 1.70
x - 4.06 = 0.935
x = 4.995

Therefore, the student with a z-score of 0.55 had a rating of approximately 4.995 for their admiration of Hillary Rodham Clinton.