Question

a set of data has a normal distribution with a mean of 55 and a standard deviation of 8. find the percent of data within the following interval
from 39 to 71

the percent of data within the given interval is
(type an integer or a decimal )

Explanation:

Step1: Calculate the number of standard deviations from the mean for the interval bounds.

First, find how many standard deviations 39 is from the mean (55) and how many standard deviations 71 is from the mean. The formula for the number of standard deviations (z - score) is \( z=\frac{x - \mu}{\sigma} \), where \( x \) is the value, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.

For \( x = 39 \):
\( z_1=\frac{39 - 55}{8}=\frac{- 16}{8}=- 2 \)

For \( x = 71 \):
\( z_2=\frac{71 - 55}{8}=\frac{16}{8}=2 \)

Step2: Use the empirical rule for normal distribution.

The empirical rule (68 - 95 - 99.7 rule) states that for a normal distribution:

• Approximately 68% of the data lies within \( \mu\pm\sigma \) (z - scores between - 1 and 1).
• Approximately 95% of the data lies within \( \mu\pm2\sigma \) (z - scores between - 2 and 2).
• Approximately 99.7% of the data lies within \( \mu\pm3\sigma \) (z - scores between - 3 and 3).

Since our z - scores are - 2 and 2, the interval \( 39 = 55-2\times8 \) to \( 71 = 55 + 2\times8 \) is \( \mu\pm2\sigma \). By the empirical rule, approximately 95% of the data lies within this interval.

Answer:

95