Science scores for HS seniors in the US are normally distributed with a mean of 77 and a standard deviation of 7. Students scoring in the top 4% are eligible for a special prize. What is the approximate cutoff score a student must get in order to receive the prize?
To find the approximate cutoff score for receiving the prize, we need to determine the value that corresponds to the top 4% of the distribution. Here's how you can calculate it:
Step 1: Find the z-score corresponding to the top 4%.
The z-score formula is given by `z = (x - mean) / standard deviation`. Rearranging the formula, we have `x = (z * standard deviation) + mean`, where `x` is the desired score we want to find.
Step 2: Find the z-score that corresponds to the top 4% using the standard normal distribution table or a calculator.
Since the distribution is normal, we know that the top 4% corresponds to the area under the curve to the right of the cutoff score. By looking up the z-score that corresponds to an area of 0.04 in the standard normal distribution table, we can find the z-score.
Step 3: Substitute the known values into the formula to find the cutoff score.
Substitute the value obtained from Step 2 into the formula `x = (z * standard deviation) + mean` to calculate the cutoff score.
Now let's go through the steps to find the cutoff score:
Step 1: Find the z-score corresponding to the top 4%.
Since we want to find the top 4%, the area to the left is 1 - 0.04 = 0.96. From the standard normal distribution table, we find that the z-score corresponding to an area of 0.96 is approximately 1.75.
Step 2: Find the z-score that corresponds to the top 4%.
We found the z-score to be approximately 1.75.
Step 3: Substitute the known values into the formula to find the cutoff score.
Using the formula `x = (z * standard deviation) + mean`, we have `x = (1.75 * 7) + 77`. Calculating this expression, the cutoff score is approximately 88.25.
Therefore, a student must score approximately 88.25 or higher to receive the special prize.