An analysis of the combined SAT scores (Verbal + Math) of a sample of psychology majors at one school indicated that the mean was 939 and the standard deviation was 166. Show your calculations used the answer the following two questions.
a. A special psychology research program only accepts applicants who had an SAT score that placed them in the top 20% of the students. What is the minimum SAT score that would place them in the top 20%?
b. Special tutors are available to assist psychology students who had an SAT score in the bottom 10% of students. What SAT score cuts off the bottom 10% of this group of students?
a. Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the Z score that fits that proportion. Insert the three values into the equation above to calculate the score.
b. Go through the same process.
Given a mean of 60, with a standard deviation of 12, and with a raw score of 75, find: (a) the z-score, (b) percentile rank, (c) t-score, (d) SAT score, and (e) stanine score. Discuss why there are so many kinds of standardized scores.
To answer these questions, we need to use the concept of z-scores. A z-score measures the number of standard deviations an observation is from the mean.
a. To find the minimum SAT score that would place students in the top 20%, we need to determine the z-score associated with the 80th percentile (since the top 20% is equivalent to the 80th percentile).
To find the z-score, we use the formula:
z = (x - μ) / σ
Where:
- x is the observation (SAT score)
- μ is the mean
- σ is the standard deviation
In this case, we need to find the SAT score (x) that corresponds to the 80th percentile, which is equivalent to finding the z-score associated with the 80th percentile.
To find the z-score associated with the 80th percentile, we use a z-table (also known as a standard normal distribution table) or calculator. The z-table provides the area under the standard normal curve (from the left) for different z-scores.
We want to find the z-score that has an area of 0.8 to the left of it. By looking up this value in the z-table, we will get the corresponding z-score.
Once we have the z-score, we can use the z-score formula mentioned earlier to find the minimum SAT score that would place students in the top 20%.
b. To find the SAT score that cuts off the bottom 10% of students, we need to determine the z-score associated with the 10th percentile.
Similarly, we will find the z-score by looking up the area of 0.1 in the z-table. Using the z-score formula, we can then find the corresponding SAT score.
Let's calculate these values using the given mean and standard deviation:
a. Calculate minimum SAT score for the top 20%:
To find the z-score for the 80th percentile, we need to find the z-score that corresponds to an area of 0.8 to the left. This area can be interpreted as 1 - 0.2 (since the top 20% is the complement of the bottom 80%).
Using a z-table or calculator, we can find that the z-score corresponding to an area of 0.8 is approximately 0.84.
Now we can substitute this value into the z-score formula:
0.84 = (x - 939) / 166
Solving for x, we have:
x - 939 = 0.84 * 166
x - 939 = 139.44
x = 939 + 139.44
x ≈ 1078.44
So the minimum SAT score that would place students in the top 20% is approximately 1078.44.
b. Calculate SAT score cutoff for the bottom 10%:
To find the z-score for the 10th percentile, we need to find the z-score that corresponds to an area of 0.1 to the left.
Using a z-table or calculator, we can find that the z-score corresponding to an area of 0.1 is approximately -1.28.
Using the z-score formula:
-1.28 = (x - 939) / 166
Solving for x, we have:
x - 939 = -1.28 * 166
x - 939 = -212.48
x = 939 - 212.48
x ≈ 726.52
So the SAT score cutoff for the bottom 10% is approximately 726.52.