Understanding the Area Under the Normal Curve
The area under the normal curve between two z-scores represents the probability of a data point falling between those two z-scores in a standard normal distribution. For the z-scores you mentioned, z = 1.0 and z = 2.0, we can calculate this using standard normal distribution tables or cumulative distribution functions.
Finding the Area
1. **Identify the Z-scores**: In this case, we have z = 1.0 and z = 2.0.
2. **Use a Z-table or calculator**: To find the area under the normal curve, you can look up the cumulative probabilities for each z-score:
- The cumulative probability for z = 1.0 is approximately 0.8413.
- The cumulative probability for z = 2.0 is approximately 0.9772.
Calculate the Area Between the Z-scores
To determine the area between z = 1.0 and z = 2.0, subtract the cumulative probability of the lower z-score from that of the higher z-score:
Area = P(Z < 2.0) - P(Z < 1.0)
Substituting the values, we get:
Area = 0.9772 – 0.8413 = 0.1359
Interpretation
This means that there is approximately a 13.59% probability that a randomly selected score from a standard normal distribution will fall between the z-scores of 1.0 and 2.0.