Given (μ) = 500, and σ = 100. What percentage of the SAT scores falls:
A) Between 500 and 600? The percentage falls 1 σ above the (μ), 68.26% / 2 = 34.13%
B) Between 400 and 600? The percentage falls 1 σ below and 1 σ above the (μ)
C) Between 500 and 700? The percentage falls 2 σ above the (μ), 95.44% /2 = 47.72%
D) Between 300 and 700? The percentage falls 2 σ below and 2 above the (μ), 95.44%
E) Above 600? 15.87%
F) Below 300? 2.28%
I got the percentage for E and F but I am not sure how to figure out how to solve for them
600 is 1 std above the mean.
300 is 3 std below the mean.
Just look up the tails in your table. Or, play around some at
http://davidmlane.com/hyperstat/z_table.html
show full soln
To calculate the percentage of SAT scores falling above or below a certain value, you can use the standard normal distribution table.
For example, to calculate the percentage of SAT scores that fall above 600, you can follow these steps:
1. Convert the score to a z-score using the formula:
z = (x - μ) / σ
where x is the value (600), μ is the mean (500), and σ is the standard deviation (100).
Substituting the values, we get:
z = (600 - 500) / 100
= 1
2. Look up the cumulative probability for the z-score in the standard normal distribution table.
The cumulative probability represents the percentage of values that fall below or equal to the given z-score.
In this case, we want the percentage of values that fall above 600, so we need to subtract the cumulative probability from 1 (100%).
From the table, the cumulative probability for z = 1 is 0.8413.
3. Subtract the cumulative probability from 1 to get the percentage above the given value.
1 - 0.8413 = 0.1587
So, the percentage of SAT scores above 600 is 15.87%.
For calculating the percentage below a certain value, you can use the same steps but instead use the cumulative probability directly from the standard normal distribution table.
For the case of SAT scores below 300, the steps would be:
1. Convert the score to a z-score:
z = (x - μ) / σ
Substituting the values, we get:
z = (300 - 500) / 100
= -2
2. Look up the cumulative probability for the z-score in the standard normal distribution table.
From the table, the cumulative probability for z = -2 is 0.0228.
So, the percentage of SAT scores below 300 is 2.28%.
To solve for the percentages in parts E and F, we need to use the concept of the standard normal distribution.
In the standard normal distribution, the mean (μ) is 0 and the standard deviation (σ) is 1. By converting the given values to z-scores using the formula (value - μ) / σ, we can find the corresponding areas under the standard normal curve using a z-table or a statistical calculator.
Let's start with part E: finding the percentage of SAT scores that fall above 600.
Step 1: Convert 600 to a z-score:
z = (600 - μ) / σ = (600 - 500) / 100 = 1
Step 2: Find the percentage for z > 1 using a z-table or calculator.
From the z-table, the area to the left of z = 1 is 0.8413, so the area to the right (above 600) is 1 - 0.8413 = 0.1587, or 15.87%.
For part F: finding the percentage of SAT scores that fall below 300.
Step 1: Convert 300 to a z-score:
z = (300 - μ) / σ = (300 - 500) / 100 = -2
Step 2: Find the percentage for z < -2 using a z-table or calculator.
From the z-table, the area to the left of z = -2 is 0.0228, or 2.28%.
Therefore, for part E, the percentage of SAT scores above 600 is 15.87%, and for part F, the percentage below 300 is 2.28%.