Suppose that the mean score of 250,000 students on the mathematics part of the SAT is 468 and the standard deviation is 104. Suppose that you make 650 on the SAT mathematics part.
a. How many standard deviations is your score above the mean? How do you do this?
b. What percentage of the students made below this score?
My answer: 96.0%
c. How many of the 250,000 students scored higher than you? The answer key says 10,000 but I got 10,025....How do you do this or is it the rounding?
d. How many of the 250,000 scored at least 800?
The answer key says 250 but I got 176...How do you do this?
(a) well, what is (650-468)/104?
650 is 1.75 std above the mean of 468
(b) looking that up in your handy dandy Z table (or at http://davidmlane.com/hyperstat/z_table.html) we find that 96% score below 650. You are correct.
How did you get (b) correct without figuring (a)?
(c) well, if 96% scored less than you, that means that 4% scored higher. 4% of 250,000 = 10,000. How did you get that extra 25?
(d) 800 is 3.19 std. P(Z>3.19) = .0007
.0007*250000 = 175
I think the answer key is wrong on this.
a. To calculate how many standard deviations your score is above the mean, you can use the formula:
z = (x - μ) / σ
Where:
z: The number of standard deviations above or below the mean
x: Your score
μ: The mean score
σ: The standard deviation
In this case, your score is 650, the mean score is 468, and the standard deviation is 104. Plugging these values into the formula:
z = (650 - 468) / 104 = 1.75
So, your score is 1.75 standard deviations above the mean.
b. To determine what percentage of students made below your score, you need to find the cumulative probability corresponding to your z-score. You can use a standard normal distribution table or a calculator with a built-in normal distribution function.
Based on the standard normal distribution table, a z-score of 1.75 corresponds to a cumulative probability of approximately 0.9599. To convert this to a percentage, multiply by 100:
0.9599 * 100 = 95.99%
So, approximately 95.99% of students made below your score.
c. To calculate the number of students who scored higher than you, you need to find the cumulative probability corresponding to your score. You can use the same approach as in part b to find the cumulative probability.
A cumulative probability of 0.9599 means that approximately 95.99% of students scored lower than you. To find the percentage of students who scored higher, subtract this percentage from 100:
100 - 95.99 = 4.01%
To calculate the number of students, multiply this percentage by the total number of students (250,000):
0.0401 * 250000 = 10,025
So, based on this calculation, approximately 10,025 students scored higher than you. It is possible that the rounding in the answer key caused a discrepancy.
d. To determine the number of students who scored at least 800, you can follow a similar approach as in part c. However, instead of using the cumulative probability, you need to use the normal distribution to find the probability corresponding to a score of 800.
Using the formula from part a:
z = (800 - 468) / 104 = 3.173
Using a standard normal distribution table or a calculator, find the cumulative probability corresponding to a z-score of 3.173.
Based on the standard normal distribution table, a z-score of 3.173 corresponds to a cumulative probability of approximately 0.9992.
To find the percentage of students who scored at least 800, subtract this probability from 1:
1 - 0.9992 = 0.0008
To calculate the number of students, multiply this probability by the total number of students (250,000):
0.0008 * 250,000 = 200
So, based on this calculation, approximately 200 students scored at least 800. It is possible that the answer key has a different calculation or rounding method.