Question

for a population that is distributed normally with a mean of 2,087 and a standard deviation of 642, use the empirical rule to determine ( p(x < 1445) ).
34%
13.5%
16%
50%

Explanation:

Step1: Calculate the difference between mean and the value

The mean \(\mu = 2087\) and the value \(x = 1445\). The difference is \(2087-1445 = 642\). Since the standard deviation \(\sigma=642\), \(x=\mu - \sigma\).

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., \(P(\mu - \sigma<X<\mu+\sigma)=0.68\)).
• The total area under the normal - distribution curve is \(1\) (or \(100\%\)).
• The area to the left of \(\mu-\sigma\) is \(\frac{1 - 0.68}{2}=0.16\) (because the normal distribution is symmetric about the mean).

Answer:

\(16\%\)