Question

question #3
for a population that is normally distributed with a mean of 12 and a standard deviation of 2, use the empirical rule to determine ( p(x > 18) ).
15%
23.5%
2.35%
.15%

Explanation:

Step1: Calculate the number of standard deviations

The formula for the number of standard deviations \(z=\frac{x - \mu}{\sigma}\), where \(\mu = 12\), \(\sigma=2\), \(x = 18\).

$$z=\frac{18 - 12}{2}=\frac{6}{2}=3$$

Step2: Apply the Empirical Rule

The Empirical Rule states that for a normal distribution:

• Approximately \(68\%\) of the data lies within \(1\) standard deviation of the mean (\(\mu\pm\sigma\))
• Approximately \(95\%\) of the data lies within \(2\) standard deviations of the mean (\(\mu\pm2\sigma\))
• Approximately \(99.7\%\) of the data lies within \(3\) standard deviations of the mean (\(\mu\pm3\sigma\))

The proportion of data within \(\mu\pm3\sigma\) is \(99.7\%\). So the proportion of data outside \(\mu\pm3\sigma\) is \(1 - 0.997=0.003\). Since the normal distribution is symmetric, the proportion of data \(P(X >\mu + 3\sigma)\) is \(\frac{1 - 0.997}{2}\)

$$P(X>18)=\frac{1 - 0.997}{2}=0.0015 = 0.15\%$$

Answer:

\(0.15\%\)