Question

1. scores on the science section of the sat are normally distributed with a mean of 400 and a standard - deviation of 50. what percentage of sat science scores are: (4 points) above 400 a. 95% above 450 b. 84% c. 50% between 400 and 500 d. 47.5% below 450 e. 16%

Explanation:

Step1: Recall properties of normal distribution

In a normal distribution, the mean divides the data symmetrically. 50% of data is above the mean and 50% is below. Since the mean of SAT science scores is 400, the percentage of scores above 400 is 50%.

Step2: Calculate z - score for 450

The z - score formula is $z=\frac{x-\mu}{\sigma}$, where $x = 450$, $\mu=400$, and $\sigma = 50$. So $z=\frac{450 - 400}{50}=\frac{50}{50}=1$.

Step3: Use z - table

Looking up the z - score of 1 in the standard normal distribution table, the area to the left of $z = 1$ is approximately 0.8413. So the percentage of scores above 450 is $1-0.8413 = 0.1587\approx16\%$.

Step4: Calculate percentage between 400 and 500

For $x = 500$, $z=\frac{500 - 400}{50}=2$. The area to the left of $z = 2$ is approximately 0.9772. The area to the left of $z = 0$ (mean) is 0.5. So the percentage between 400 and 500 is $0.9772-0.5 = 0.4772\approx47.5\%$.

Step5: Calculate percentage below 450

Since the area to the left of $z = 1$ (for $x = 450$) is approximately 0.8413, the percentage of scores below 450 is 84%.

Answer:

above 400: c. 50%
above 450: e. 16%
between 400 and 500: d. 47.5%
below 450: b. 84%