The scores on an exam are normally distributed, with a mean of 77 and a standard deviation of 10. What percent of the scores are greater than 87?
* 68%
* 16%
* 84%
* 2.5%
16%
Well, if the scores on the exam were too high, they might just float away! But to answer your question, we need to find the area under the normal distribution curve that represents scores greater than 87. Since we know the mean is 77 and the standard deviation is 10, we can use the Z-score formula:
Z = (X - μ) / σ
where X is the value we're interested in (87), μ is the mean (77), and σ is the standard deviation (10).
Plugging in the numbers:
Z = (87 - 77) / 10
Z = 10 / 10
Z = 1
Now we need to find the area to the right of Z = 1. We can use a Z-table or a calculator to find this area. Looking up Z = 1 in the Z-table, we find that the area to the right is approximately 0.1587.
To find the percentage, we need to multiply this decimal by 100:
0.1587 * 100 = 15.87%
So, approximately 15.87% of the scores are greater than 87. But don't worry, that's just a reminder that we shouldn't aim too high, unless we're trying to brush the clouds with our pencils!
To find the percent of scores that are greater than 87, we need to determine the area under the normal distribution curve to the right of 87.
Step 1: Calculate the z-score.
The z-score formula is given by:
z = (x - μ) / σ
Where:
x = individual score (87 in this case)
μ = mean (77)
σ = standard deviation (10)
Plugging in the values:
z = (87 - 77) / 10
z = 10 / 10
z = 1
Step 2: Determine the area under the curve.
Using a standard normal distribution table or calculator, we find that the area to the right of 1 corresponds to approximately 0.1587.
Step 3: Convert the area to a percentage.
To convert the area to a percentage, we multiply by 100:
0.1587 * 100 = 15.87%
Therefore, approximately 15.87% of the scores are greater than 87.
None of the given answer choices (68%, 16%, 84%, 2.5%) match the correct percentage.
To find the percentage of scores that are greater than 87, we need to use the properties of a normal distribution.
First, we need to calculate the Z-score for 87. The Z-score measures how many standard deviations a particular value is away from the mean. The formula for calculating the Z-score is:
Z = (X - μ) / σ
Where:
X = Value for which we want to calculate the Z-score (in this case, 87)
μ = Mean of the distribution (77)
σ = Standard deviation of the distribution (10)
Using these values in the formula, we can calculate the Z-score for 87:
Z = (87 - 77) / 10 = 10 / 10 = 1
Next, we need to find the percentage of scores that are greater than a Z-score of 1. We can use a standard normal distribution table or a calculator to find this percentage.
Looking up the Z-score of 1 in the standard normal distribution table, we find that the area to the left of Z = 1 is approximately 0.8413.
Since we want the percentage that is greater than 1, we subtract this value from 1:
1 - 0.8413 = 0.1587
So, approximately 15.87% of the scores are greater than 87.
Therefore, the correct answer is:
- 16%