Assume that the math SAT scores are normally distributed with a mean of 500 and a standard deviation of 100. If you score 560 on this exam, what percentage of those taking the test scored below you?
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Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to the Z score. Hint: in smaller portion.
To find the percentage of test-takers who scored below 560 on the math SAT, we need to use the concept of z-scores and the standard normal distribution table.
The z-score is a measure of how many standard deviations above or below the mean a particular data point is. In this case, we need to calculate the z-score for a score of 560.
The formula to calculate the z-score is:
z = (x - μ) / σ
Where:
- x represents the given score (560 in this case)
- μ represents the mean (500 in this case)
- σ represents the standard deviation (100 in this case)
Using these values, we can calculate the z-score:
z = (560 - 500) / 100
z = 60 / 100
z = 0.6
Now, we need to find the proportion of the area under the normal distribution curve that lies to the left of this z-score. This proportion represents the percentage of test-takers who scored below 560.
We can use a standard normal distribution table to find this proportion. For a z-score of 0.6, the table will give us the corresponding cumulative probability. In this case, the cumulative probability is 0.7257.
Multiplying this probability by 100 will give us the percentage:
Percentage = 0.7257 * 100 = 72.57%
Therefore, approximately 72.57% of test-takers scored below 560 on the math SAT.