Question

question #1
for a population that is distributed normally with a mean of 100 and a standard deviation of 15, use the empirical rule to determine
p(100 < x < 130).

97.5%
47.5%
16%
84%

question #2
for a population that is distributed normally with a mean of 37 and a standard deviation of 3, use the empirical rule to determine
p(x > 31).

13.5%
97.5%
84%
16%

Explanation:

Question #1

Step1: Calculate the number of standard deviations

The mean \(\mu = 100\) and the standard deviation \(\sigma=15\). For \(x = 130\), the number of standard deviations from the mean is \(z=\frac{130 - 100}{15}=\frac{30}{15} = 2\)

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 normal distribution is symmetric about the mean. The probability \(P(\mu - 2\sigma<X<\mu + 2\sigma)=0.95\). Since the distribution is symmetric about \(x = \mu\), \(P(\mu<X<\mu + 2\sigma)=\frac{0.95}{2}=0.475 = 47.5\%\)

Question #2

Step1: Calculate the number of standard deviations

The mean \(\mu = 37\) and the standard deviation \(\sigma = 3\). For \(x = 31\), the number of standard deviations from the mean is \(z=\frac{31-37}{3}=\frac{-6}{3}=- 2\)

Step2: Apply the Empirical Rule

We know that \(P(\mu - 2\sigma<X<\mu + 2\sigma)=0.95\). Then \(P(X<\mu - 2\sigma)=\frac{1 - 0.95}{2}=0.025\)

Using the complement rule \(P(X>31)=1 - P(X\leq31)\). Since \(P(X\leq31)=P(X<\mu - 2\sigma) = 0.025\), then \(P(X>31)=1-0.025 = 0.975=97.5\%\)

Answer:

Question #1: \(47.5\%\)
Question #2: \(97.5\%\)