1. A distribution has a standard deviation of � � 12.

Find the z-score for each of the following locations
in the distribution.
a. Above the mean by 3 points.
b. Above the mean by 12 points.
c. Below the mean by 24 points.
d. Below the mean by 18 points.

For the following population of N � 6 scores:
3, 1, 4, 3, 3, 4
a. Sketch a histogram showing the population
distribution.
b. Locate the value of the population mean in your
sketch, and make an estimate of the standard
deviation (as done in Example 4.2).
c. Compute SS, variance, and standard deviation for
the population. (How well does your estimate
compare with the actual value of �?)

1. Z = (score-mean)/SD

a. Z = 3/12
You can calculate the rest.

2. a. We cannot sketch a Histogram on a post.

b, c. 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

I'll let you do the calculations.

.25

To find the z-score for each of the given locations in the distribution with a standard deviation of 12, we can use the formula:

z = (x - μ) / σ

Where:
z = z-score
x = value in the distribution
μ = mean of the distribution
σ = standard deviation of the distribution

Let's calculate the z-scores for each of the given locations:

a. Above the mean by 3 points.
x = μ + 3
z = (μ + 3 - μ) / 12 = 3 / 12 = 0.25

b. Above the mean by 12 points.
x = μ + 12
z = (μ + 12 - μ) / 12 = 12 / 12 = 1

c. Below the mean by 24 points.
x = μ - 24
z = (μ - 24 - μ) / 12 = -24 / 12 = -2

d. Below the mean by 18 points.
x = μ - 18
z = (μ - 18 - μ) / 12 = -18 / 12 = -1.5

So the z-scores for the given locations in the distribution are:
a. 0.25
b. 1
c. -2
d. -1.5

Now let's move on to the next part of the question.

For the population of N = 6 scores: 3, 1, 4, 3, 3, 4

a. Sketch a histogram showing the population distribution.

[Histogram Sketch]

b. Locate the value of the population mean in your sketch, and make an estimate of the standard deviation (as done in Example 4.2).

From the histogram, it appears that the mean is around 3. Take note of where the peak of the distribution is located.

As for the estimate of the standard deviation, we can see that the scores are relatively close together, so we can assume that the standard deviation is relatively small, perhaps around 1-2.

c. Compute SS, variance, and standard deviation for the population. (How well does your estimate compare with the actual value of σ?)

First, let's calculate the sum of squares (SS):

SS = (3 - μ)² + (1 - μ)² + (4 - μ)² + (3 - μ)² + (3 - μ)² + (4 - μ)²

Next, let's calculate the variance:

variance = SS / N = [sum of squares] / N

Finally, the standard deviation is the square root of the variance:

standard deviation = √(variance)

Since we do not have the actual value of μ, we cannot calculate the exact values of SS, variance, and standard deviation. But you can substitute the estimated value of μ from step b into the formulas to get an approximate calculation.

To get the z-score for each of the given locations in the distribution (with a standard deviation of 12), you can use the z-score formula:

z = (x - μ) / σ

Where:
- z is the z-score
- x is the value in the distribution
- μ is the mean of the distribution
- σ is the standard deviation of the distribution

a. Above the mean by 3 points:
To calculate the z-score, we need the value of the mean, but it is not provided in the question. Assuming the mean is 0, the calculation would be:
z = (0 + 3) / 12 = 3/12 = 0.25

b. Above the mean by 12 points:
Using the same assumption of a mean of 0:
z = (0 + 12) / 12 = 12/12 = 1

c. Below the mean by 24 points:
Assuming the mean is 0:
z = (0 - 24) / 12 = -24/12 = -2

d. Below the mean by 18 points:
Assuming the mean is 0:
z = (0 - 18) / 12 = -18/12 = -1.5

Now, let's move on to the next question.

a. Sketch a histogram showing the population distribution:
To construct the histogram, you can plot the values on the x-axis and the frequency on the y-axis. For the given population, the histogram would look like this:

Frequency
| x
| x x
| x x
|_____
0 1 2 3 4

b. Locate the value of the population mean in your sketch and make an estimate of the standard deviation:
In the histogram, the mean can be approximated as the value around which the distribution is centered. Based on the sketch, it appears that the mean is close to 3. For the estimate of the standard deviation, you can visually assess the spread of the distribution. It seems to be around 1.

c. Compute SS, variance, and standard deviation for the population:
To calculate the sum of squares (SS), you need to calculate the squared deviation from the mean for each value, then add them up:
SS = (3-3)^2 + (1-3)^2 + (4-3)^2 + (3-3)^2 + (3-3)^2 + (4-3)^2 = 2

To find the variance, divide the sum of squares by the number of values (N):
variance = SS / N = 2 / 6 = 1/3

Finally, to calculate the standard deviation, take the square root of the variance:
standard deviation = √(variance) = √(1/3) ≈ 0.58

This estimation of the standard deviation may not be accurate since it was based on a visual assessment. However, it gives an approximation of the spread of the data in the population.