Final answer:
To find the standard deviation of a normally distributed data set with a mean of 31 and 95% of data between 27.4 and 34.6, we use the empirical rule. The standard deviation is calculated to be 1.8 by halving the difference between the mean and the endpoints of the range.
Step-by-step explanation:
The question deals with determining the standard deviation of a normally distributed data set with known mean and a range that encloses 95% of the data. According to the empirical rule, 95% of data in a normal distribution falls within two standard deviations (2σ) of the mean. Given a mean of 31 and a range from 27.4 to 34.6, we can calculate the standard deviation as follows:
Upper boundary corresponding to mean + 2σ = 31 + 2σ = 34.6
Lower boundary corresponding to mean - 2σ = 31 - 2σ = 27.4
Solving these equations gives us 2 standard deviations:
2σ = 34.6 - 31 = 3.6
2σ = 31 - 27.4 = 3.6
Now, to find the standard deviation (σ), we divide the value of 2σ by 2:
σ = 3.6 / 2 = 1.8
Therefore, the standard deviation of the data set is 1.8