The area under the standard normal distribution curve between two z-scores represents the probability that a value falls within that range. To find this area, we can utilize the standard normal distribution table (z-table) or software that calculates the cumulative distribution function (CDF).
1. **Understanding the Z-Scores:**
In this case, we have two z-scores: z = 1.50 and z = 2.50. These scores are standard deviations from the mean of the distribution, which is 0 for the standard normal distribution.
2. **Looking Up the Z-Scores:**
We first look up the cumulative probability for each z-score in the z-table:
– For z = 1.50, the table shows that the cumulative area is approximately 0.9332.
– For z = 2.50, the cumulative area is approximately 0.9938.
3. **Calculating the Area Between the Z-Scores:**
To find the area between z = 1.50 and z = 2.50, we subtract the cumulative probability of the lower z-score from that of the higher z-score:
Area = P(Z < 2.50) - P(Z < 1.50)
Area = 0.9938 – 0.9332 = 0.0606
4. **Interpreting the Result:**
This means that the probability of a value falling between z = 1.50 and z = 2.50 is approximately 0.0606, or 6.06%. Therefore, if you were to randomly select a value from a standard normal distribution, there’s about a 6.06% chance that it would fall between these two z-scores.
In summary, the area under the standard normal distribution curve between z = 1.50 and z = 2.50 is approximately 0.0606.