(population)
A normal distribution has a mean of 80 and a std dev of 20. For each of the following scores, indicate wheter the tail of the distribution is located to the right or left of the score, and find the proportion of the distribution located in the tail.
a.X=85, B.X=92, C. X=70 D. X=-64
If the score is above the mean (ƒÊ), the tail will be to the right — and to the left, if below the mean.
The exact proportions can be found by using the Z score formula
Z = (x - ƒÊ)/SD
and looking those Z values up in a table in the back of your statistics text called something like "areas under the normal distribution."
I hope this helps. Thanks for asking.
To determine whether the tail of the distribution is located to the right or left of a score and to find the proportion of the distribution located in the tail, we need to compare the scores to the mean.
In a normal distribution, the mean is the center point of the distribution. The standard deviation indicates the spread of the data around the mean.
Let's calculate the z-score for each of the given scores using the formula:
z = (X - mean) / std dev
a. X = 85:
Z = (85 - 80) / 20 = 0.25
Since the z-score is positive, the tail of the distribution is located to the right of the score. To find the proportion of the distribution located in the tail, we need to calculate the area under the curve to the right of the z-score. We can look up this area in a Z-table or use statistical software.
Looking up the area in a Z-table, we find that the proportion of the distribution located in the tail is approximately 0.4013.
b. X = 92:
Z = (92 - 80) / 20 = 0.60
Similar to case a, since the z-score is positive, the tail of the distribution is located to the right of the score. By looking up the area in the Z-table, we find that the proportion of the distribution located in the tail is approximately 0.2743.
c. X = 70:
Z = (70 - 80) / 20 = -0.50
This time, the z-score is negative, indicating that the tail of the distribution is located to the left of the score. By looking up the area in the Z-table for the corresponding positive z-score (0.50), we can find the proportion of the distribution located in the tail. Subtracting this area from 0.5 (since we are looking for the left tail), we find that the proportion of the distribution located in the tail is approximately 0.3085.
d. X = -64:
Z = (-64 - 80) / 20 = -7
The negative z-score indicates that the tail of the distribution is located to the left of the score. By looking up the area in the Z-table for the corresponding positive z-score (7), and subtracting it from 0.5, we find that the proportion of the distribution located in the tail is approximately 0.0000.
Keep in mind that the proportions obtained from the Z-table are approximate values. For more accurate calculations, statistical software can be used.