Suppose 95% of the data fall between 1.4 and 46.8. Using the empirical rule, give an estimate of the standard deviation.
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The empirical rule, also known as the 68-95-99.7 rule, states that for a normally distributed data set, approximately:
- 68% of the data falls within one standard deviation of the mean,
- 95% falls within two standard deviations, and
- 99.7% falls within three standard deviations.
In your case, you have been given that 95% of the data falls between 1.4 and 46.8. Since this interval is two standard deviations from the mean, we can use this information to estimate the standard deviation.
To find the standard deviation, we need to determine the distance between the mean and each of the two boundary values. Since 95% falls within two standard deviations, we can calculate the distance by dividing the range (difference between the two boundary values) by four.
The range can be found by subtracting the lower boundary from the upper boundary:
Range = 46.8 - 1.4 = 45.4
Now, divide the range by four to get the estimate of the standard deviation:
Standard Deviation = Range / 4 = 45.4 / 4 = 11.35
Therefore, using the empirical rule, the estimate of the standard deviation is 11.35.