Question

question #1
for a population that is distributed normally with a mean of 45 and a standard deviation of 5, use the empirical rule to determine ( p(x < 50) ).

97.5%
13.5%
16%
84%

Explanation:

Step1: Calculate the number of standard deviations from the mean

The mean \(\mu = 45\) and the standard deviation \(\sigma=5\). For \(x = 50\), the \(z - \)score is \(z=\frac{x-\mu}{\sigma}=\frac{50 - 45}{5}=1\)

Step2: Apply the Empirical Rule

The Empirical Rule states that for a normal distribution:

• Approximately \(68\%\) of the data lies within \(\mu\pm\sigma\) (i.e., \(z=- 1\) to \(z = 1\))
• Approximately \(95\%\) of the data lies within \(\mu\pm2\sigma\)
• Approximately \(99.7\%\) of the data lies within \(\mu\pm3\sigma\)

The total area under the normal curve is \(100\%\). The area to the left of \(z = 1\) is composed of the area within \(\mu-\sigma\) to \(\mu+\sigma\) (\(68\%\)) and the area to the left of \(\mu-\sigma\) (\(16\%\)) plus the area from \(\mu\) to \(\mu+\sigma\) (\(34\%\)). So \(P(X<50)=P(Z < 1)=50\%+34\%=84\%\)

Answer:

\(84\%\) (corresponding to the fourth option)