Scores on the SAT form a normal distribution with u=500 and o=100.
a. You are trying to qualify for a scholarship and find out that you need to have a higher score than at least 75% of the students. What score will you need?
b. What is the minimum score necessary to be in the top 15% of the SAT distribution?
Please show me how to work these problems.
I came up with a) 559.87 but I don't think I did it correctly (0.5987 *100+500). I can't even figure b.
To solve these problems, we will use the concept of Z-scores and the standard normal distribution table. The Z-score measures the number of standard deviations a data point is from the mean of a distribution.
a. To find the score you need to be higher than at least 75% of the students:
1. Identify the Z-score corresponding to the desired percentile. In this case, we need to find the Z-score for the 75th percentile.
2. Look up the Z-score in the standard normal distribution table to find its corresponding value. The closest value to 75% in the table is 0.674.
3. Use the formula Z = (X - μ) / σ, where X is the desired score, μ is the mean, and σ is the standard deviation of the distribution.
Rearranging the formula, we have X = Z * σ + μ.
4. Plug in the known values: Z = 0.674, μ = 500, and σ = 100.
Calculate X: X = 0.674 * 100 + 500 = 673.4.
Rounding to the nearest whole number, you will need a score of approximately 673.
b. To find the minimum score necessary to be in the top 15% of the SAT distribution:
1. Identify the Z-score corresponding to the desired percentile, which is 85% since we want the top 15%.
2. Look up the Z-score in the standard normal distribution table to find its corresponding value. The value closest to 85% is 1.036.
3. Use the formula X = Z * σ + μ.
4. Plug in the known values: Z = 1.036, μ = 500, and σ = 100.
Calculate X: X = 1.036 * 100 + 500 = 613.6.
Rounding to the nearest whole number, the minimum score necessary to be in the top 15% is approximately 614.
So, the correct answers are:
a) You will need a score of approximately 673.
b) The minimum score necessary to be in the top 15% is approximately 614.