SAT math scores approx. follow a normal distribution, with mean 500 and standard deviation 100. What % of students should have an SAT math score above 500?
I think I use mean (500) + z-score * SD (100), but I do not know how to get the z-score?
You don't need a z-score for this problem. A normal distribution means that half the observations are above the mean, half below. The answer is 50%
To find the z-score, you can use the formula:
z = (x - μ) / σ
where
- z is the z-score
- x is the value you want to find the z-score for (in this case, 500)
- μ is the mean of the distribution (500 in this case)
- σ is the standard deviation of the distribution (100 in this case)
In this particular scenario, since you want to find the percentage of students who scored above 500, you need to find the area under the curve to the right of 500.
Since the distribution is symmetric, the z-score for the 50th percentile (the mean) is 0. The area under the curve to the right of 500 would be the same as the area to the left of negative infinity (which is 0.5 on a standard normal distribution). Therefore, the percentage of students who scored above 500 would be 1 - 0.5 = 0.5 or 50%.
In conclusion, 50% of students should have an SAT math score above 500.
To find the percentage of students with an SAT math score above 500, you need to determine the z-score and then use the standard normal table.
The z-score tells you how many standard deviations above or below the mean a particular value is. To find the z-score, you can use the formula:
z = (x - μ) / σ
where:
- x is the value for which you want to find the z-score (in this case, 500)
- μ is the mean of the distribution (500)
- σ is the standard deviation of the distribution (100)
Substituting the values into the formula, we get:
z = (500 - 500) / 100
= 0 / 100
= 0
Since the z-score is 0, this means that a score of 500 is at the mean of the distribution.
In a standard normal distribution, which has a mean of 0 and a standard deviation of 1, we can use the standard normal table to find the percentage.
Since our z-score is 0, we can look up the probability associated with a z-score of 0 in the standard normal table. The table value for a z-score of 0 is 0.5000, which represents 50%.
This means that approximately 50% of students should have an SAT math score above 500, assuming the scores follow a normal distribution with a mean of 500 and a standard deviation of 100.