Question

scores on a certain standardized test have a mean of 500, and a standard deviation of 100. how common is a score between 600 and 700? calculate the probability.

Explanation:

Step1: Identify the distribution

The scores follow a normal distribution (implied by standardized test and mean, standard deviation given) with mean $\mu = 500$ and standard deviation $\sigma = 100$. We need to find $P(600 < X < 700)$.

Step2: Calculate z - scores

The formula for z - score is $z=\frac{X-\mu}{\sigma}$.
For $X = 600$: $z_1=\frac{600 - 500}{100}=\frac{100}{100}=1$
For $X = 700$: $z_2=\frac{700 - 500}{100}=\frac{200}{100}=2$

Step3: Use the empirical rule

The empirical rule for normal distribution states that:

• $P(\mu-\sigma
• $P(\mu - 2\sigma
• $P(\mu - 3\sigma

We know that $P(500

To find $P(600

$P(500 < X<700)\approx47.5\%$ and $P(500 < X<600)\approx34\%$

So $P(600 < X<700)=47.5\% - 34\%=13.5\%$

Answer:

13.5%