Express the interval using the standard deviation.
99.7 %
Practice makes perfect
We have been told that the masses of pennies minted in the U.S. after 1982 are normally distributed with a mean of 2.50g and a standard deviation of 0.02g. We will find the percentages of pennies that have a mass between 2.44g and 2.56g.
2.44< x < 2.56
First, let's find the difference between the mean, 2.50g, and the lower limit, 2.44g, in terms of standard deviations. To do that, we will subtract the lower limit from the mean.
2.50g- 2.44g= 0.06 g
Next, we will divide the difference by the standard deviation, 0.02g.
0.06 g/0.02g=3
This means that 2.44 is 3 standard deviations below the mean. We will do the same thing for the upper limit. This time, we will subtract the mean, 2.50g, from the upper limit, 2.56g.
2.56g- 2.50g= 0.06 g
Then, we will divide the difference by the standard deviation, 0.02g.
0.06 g/0.02g=3
The upper limit is also 3 standard deviations above the mean. As a result, the masses of the pennies between 2.44g and 2.56g fall within 3 standard deviations of the mean.
Normally distributed data always contains approximately 99.7 % of the data within 3 standard deviations on both sides.
