Question

x is a normally distributed random variable with mean 28 and standard deviation 10. what is the probability that x is between 18 and 58? use the 0.68 - 0.95 - 0.997 rule and write your answer as a decimal. round to the nearest thousandth if necessary.

Explanation:

Step1: Calculate z - scores

For \(x = 18\), \(z_1=\frac{18 - 28}{10}=\frac{- 10}{10}=-1\). For \(x = 58\), \(z_2=\frac{58 - 28}{10}=\frac{30}{10}=3\).

Step2: Apply the 68 - 95 - 99.7 rule

The 68 - 95 - 99.7 rule states that about 68% of the data is within 1 standard - deviation of the mean (\(z=\pm1\)), about 95% is within 2 standard - deviations of the mean (\(z = \pm2\)), and about 99.7% is within 3 standard - deviations of the mean (\(z=\pm3\)). The area between \(z=-1\) and \(z = 3\) can be found as follows: The area between \(z=-3\) and \(z = 3\) is approximately 0.997, and the area between \(z=-3\) and \(z=-1\) is half of the area outside of \(z=-1\) and \(z = 1\). The area outside of \(z=-1\) and \(z = 1\) is \(1 - 0.68=0.32\), so the area between \(z=-3\) and \(z=-1\) is \(\frac{0.32}{2}=0.16\). Then the area between \(z=-1\) and \(z = 3\) is \(0.997-0.16 = 0.837\).

Answer:

0.837