Suppose 95% of the data fall between 12.4 and 46.8. Using the empirical rule, give an estimae of the standard deviation
To estimate the standard deviation using the empirical rule, we can make use of the fact that for data following a normal distribution, about 68% of the data falls within one standard deviation of the mean, about 95% falls within two standard deviations, and about 99.7% falls within three standard deviations.
Here, we are given that 95% of the data falls between 12.4 and 46.8. To estimate the standard deviation, we can start by considering the range of two standard deviations.
We can calculate this by finding the distance between the two values where 95% of the data falls.
The range of two standard deviations is obtained by multiplying the difference between the upper and lower bounds by 2. In other words:
Range of Two Standard Deviations = 2 * (Upper Bound - Lower Bound)
In this case, the upper bound is 46.8 and the lower bound is 12.4, so we have:
Range of Two Standard Deviations = 2 * (46.8 - 12.4) = 2 * 34.4 = 68.8
Since we know that 95% of the data falls within two standard deviations, the range of two standard deviations represents 95% of the data. Therefore, we can estimate that the standard deviation is approximately equal to one-half of the range of two standard deviations:
Standard Deviation ≈ (Range of Two Standard Deviations) / 2
Standard Deviation ≈ 68.8 / 2 = 34.4
So, using the empirical rule, we can estimate the standard deviation to be approximately 34.4.