Question

assume that a normal distribution has a mean of 22 and a standard deviation of 4. what percentage of the values in the distribution do we expect to fall between 18 and 22?

a. 25%
b. 17%
c. 34%
d. 68%

Explanation:

Step1: Calculate the z - score

The formula for the z - score is \(z=\frac{x-\mu}{\sigma}\), where \(x\) is the value, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.
For \(x = 18\), \(\mu=22\), \(\sigma = 4\), then \(z=\frac{18 - 22}{4}=\frac{-4}{4}=-1\)
For \(x = 22\), \(z=\frac{22-22}{4}=0\)

Step2: Use the empirical rule (68 - 95 - 99.7 rule)

The empirical rule states that for a normal distribution:

• Approximately 68% of the data lies within \(z=-1\) and \(z = 1\)
• Approximately 95% of the data lies within \(z=-2\) and \(z = 2\)
• Approximately 99.7% of the data lies within \(z=-3\) and \(z = 3\)

The percentage of data between \(z=-1\) and \(z = 0\) is half of the percentage of data between \(z=-1\) and \(z = 1\). Since the percentage of data between \(z=-1\) and \(z = 1\) is 68%, the percentage of data between \(z=-1\) and \(z = 0\) is \(\frac{68\%}{2}=34\%\)

Answer:

C. 34%