Final answer:
To calculate the standard deviation of the normally distributed data set with a mean of 31 and 95% of the data between 27.4 and 34.6, we use the formula for two standard deviations from the mean. The calculation shows that the standard deviation is 1.8. The correct answer is option c.
Step-by-step explanation:
The given problem is associated with the properties of normal distribution. We have a data set with a mean (μ) of 31 and 95% of the data falls between 27.4 and 34.6. The range of values within which 95% of the data falls corresponds to two standard deviations on either side of the mean in a normal distribution according to the empirical rule.
To find the standard deviation, we can subtract the mean from the higher value (or lower value) and then divide by 2. Hence:
μ + 2σ = 34.6
μ - 2σ = 27.4
Now we solve for σ (standard deviation) using:
σ = (34.6 - 31) / 2 = 3.6 / 2 = 1.8
Therefore, the correct option for the standard deviation of the data set is c. 1.8.