Normal curve 1.5 standard deviations

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For a normal curve, how much of the area lies within 1.5 standard deviations of the mean? I already know about the 68–95–99.7 rule, and see that it should be between 68% and 95%. I also know that it should be closer to 95%, so I estimate it to be around 80%. Can someone give me a hint on how to find the answer?

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You're close. It's about 87%. And see: probit.