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: Calculate z - scores

The formula for the z - score is \(z=\frac{x-\mu}{\sigma}\), where \(\mu = 500\) (mean) and \(\sigma=100\) (standard deviation).
For \(x = 600\): \(z_1=\frac{600 - 500}{100}=1\)
For \(x = 700\): \(z_2=\frac{700 - 500}{100}=2\)

Step2: Use the empirical rule (68 - 95 - 99.7 rule)

The empirical rule states that for a normal distribution:

• Approximately \(68\%\) of the data lies within \(z=\pm1\)
• Approximately \(95\%\) of the data lies within \(z = \pm2\)
• Approximately \(99.7\%\) of the data lies within \(z=\pm3\)

The proportion of data between \(z = 1\) and \(z = 2\) is \(\frac{95\% - 68\%}{2}\)

$$ \frac{95 - 68}{2}\%=\frac{27}{2}\% = 13.5\% $$

Answer:

\(13.5\%\)