Answer
$P(-1\lt z\lt 1)=0.6827$
$P(-2\lt z\lt 2)=0.9545$
$P(-3\lt z\lt 3)=0.9973$
see comparisons below.
Work Step by Step
The empirical rule states that
• Approximately 68% of the data values will fall within 1 standard deviation of the mean.
• Approximately 95% of the data values will fall within 2 standard deviations of the mean.
• Approximately 99.7% of the data values will fall within 3 standard deviations of the mean.
$P(-1\lt z\lt 1)=1-2P(Z=-1)=0.6827$ while the empirical rule gives 0.68 which is fairly close.
$P(-2\lt z\lt 2)=1-2P(Z=-2)=0.9545$ while the empirical rule gives 0.95 which is reasonably close.
$P(-3\lt z\lt 3)=1-2P(Z=-3)=0.9973$ while the empirical rule gives 0.997 which is very close.
