Question

question #2
a normally distributed population has a mean of 87 and a standard deviation of 23.
determine the quartiles.
o q1: 71.49 q3: 102.51
o q1: 73.24 q3: 100.76
o q1: 67.41 q3: 105.59
o q1: 69.43 q3: 103.57

question #3
for a population that is normally distributed with a mean of 10 and a standard
deviation of 2, use the empirical rule to determine p(x > 16).
o 15%
o.15%
o 23.5%
o 2.35%

Explanation:

Step1: Find the z - scores for quartiles

The first quartile \(Q_1\) has an area of \(0.25\) to the left, and the third quartile \(Q_3\) has an area of \(0.75\) to the left. Using a standard normal table or calculator, the z - score for \(Q_1\) (\(z_{0.25}\)) is approximately \(- 0.674\) and for \(Q_3\) (\(z_{0.75}\)) is approximately \(0.674\).

Step2: Use the z - score formula \(x=\mu+z\sigma\)

Given \(\mu = 87\) and \(\sigma=23\).
For \(Q_1\): \(x_1=\mu + z_{0.25}\sigma=87+( - 0.674)\times23=87 - 15.502 = 71.498\approx71.49\)
For \(Q_3\): \(x_3=\mu+z_{0.75}\sigma=87 + 0.674\times23=87+15.502 = 102.502\approx102.51\)

for Question 3:

Step1: Calculate the number of standard deviations

The formula for the z - score is \(z=\frac{x-\mu}{\sigma}\). Given \(\mu = 10\), \(\sigma = 2\) and \(x = 16\). Then \(z=\frac{16 - 10}{2}=\frac{6}{2}=3\)

Step2: Apply the Empirical Rule

The Empirical Rule states that for a normal distribution:

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

The area within \(\mu\pm3\sigma\) is \(0.997\). The area outside of \(\mu + 3\sigma\) (since the distribution is symmetric) is \(\frac{1 - 0.997}{2}=0.0015 = 0.15\%\)

Answer:

Q1: 71.49, Q3: 102.51