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. To calculate how many standard deviations your score is above the mean, you can use the formula:
Z-score = (x - mean) / standard deviation
Where:
- x is your score
- mean is the mean score of the population
- standard deviation is the standard deviation of the population
In this case, your score (x) is 650, the mean score is 468, and the standard deviation is 104.
Z-score = (650 - 468) / 104
Z-score = 182 / 104
Z-score ≈ 1.75
So, your score is approximately 1.75 standard deviations above the mean.
b. To calculate the percentage of students who made below your score, you can use the standard normal distribution table. This table provides the percentage of values below a given Z-score.
From the table, the Z-score of 1.75 corresponds to a cumulative probability of approximately 0.9599. To determine the percentage below your score, subtract this probability from 1 and convert it to a percentage:
Percentage below your score = (1 - 0.9599) * 100
Percentage below your score ≈ 4.01%
So, approximately 4.01% of students made below your score.
c. To determine the number of students who scored higher than you, you need to calculate the Z-score for your score and then convert it into a percentile.
Z-score = (x - mean) / standard deviation
Using the same formula as question a:
Z-score = (650 - 468) / 104
Z-score ≈ 1.75
From the standard normal distribution table, a Z-score of 1.75 corresponds to a percentile of approximately 0.9599.
So, the percentage of students who scored higher than you is (1 - 0.9599) * 100 ≈ 4.01%.
To find the number of students who scored higher, calculate this percentage as a fraction of the total number of students:
Number of students = Percentage * Total number of students
Number of students = 0.0401 * 250,000
Number of students ≈ 10,025
So, approximately 10,025 students scored higher than you. It seems there might be a discrepancy with the rounding.
d. To find the number of students who scored at least 800, you need to calculate the Z-score for this score and then determine the percentage below.
Z-score = (x - mean) / standard deviation
Using the formula:
Z-score = (800 - 468) / 104
Z-score ≈ 3.19
From the standard normal distribution table, a Z-score of 3.19 corresponds to a cumulative probability of approximately 0.9996.
To determine the percentage below this score, convert the cumulative probability:
Percentage below the score = 0.9996 * 100
Percentage below the score ≈ 99.96%
So, approximately 99.96% of students scored below 800.
To find the number of students who scored at least 800, subtract this percentage from 100% and calculate the fraction of the total number of students:
Number of students = (1 - 0.9996) * Total number of students
Number of students ≈ 0.0034 * 250,000
Number of students ≈ 850
So, approximately 850 students scored at least 800. There seems to be a discrepancy with the answer key.