Answer :
Final answer:
The area under the standard normal curve between z = -3 and z = +2 is 97.35%, calculated using the 68-95-99.7 rule by subtracting the areas outside of these z-scores from 100%.
Explanation:
Using the 68-95-99.7 rule, also known as the empirical rule, we can determine the area under the standard normal curve between the z-scores of -3 and +2. The rule tells us that:
- About 68% of the data falls within one standard deviation of the mean (z-scores -1 to +1).
- About 95% of the data falls within two standard deviations of the mean (z-scores -2 to +2).
- About 99.7% of the data falls within three standard deviations of the mean (z-scores -3 to +3).
To find the area between z = -3 and z = +2, we know that 95% of the data is between z = -2 and z = +2. Since about 99.7% of the data lies between z = -3 and z = +3, we can calculate this by subtracting the area outside of z = -3 and z = +2 from 100%. The area outside of z = +2 (above +2 standard deviations) is about 2.5% of the total area, so the area to the right is about 2.5%. From z = -3 to z = -2, we have about half of the remaining 0.3%, which is 0.15%. Therefore, the area between z = -3 and z = +2 under the standard normal curve is given by 100% - 2.5% - 0.15% = 97.35%.
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