The area under the standard normal curve between z = 0 and z = 3 can be found using statistical tools and properties of the normal distribution. The standard normal distribution is a symmetrical bell-shaped curve centered at a mean of 0 and a standard deviation of 1.
To find the area between the z-scores of 0 and 3, you can refer to a standard normal distribution table (Z-table) which provides the cumulative probability associated with a given z-score. The cumulative probability tells you the area under the curve to the left of that particular z-score.
1. **Find the cumulative probability for z = 0:**
– The cumulative probability at z = 0 is 0.5000 because it is the midpoint of the curve, meaning 50% of the data lies to the left of this z-score.
2. **Find the cumulative probability for z = 3:**
– Consult the Z-table or use statistical software/calculator. For z = 3, the cumulative probability is approximately 0.9987. This indicates that about 99.87% of the data lies to the left of z = 3.
3. **Calculate the area between z = 0 and z = 3:**
– To find the area between these two z-scores, simply subtract the cumulative probability at z = 0 from that at z = 3:
Area = P(Z ≤ 3) – P(Z ≤ 0) = 0.9987 – 0.5000 = 0.4987.
So, the area under the standard normal curve between z = 0 and z = 3 is approximately 0.4987 or 49.87%. This means that roughly half of the data under the curve lies between these two points, indicating a significant portion of the standard normal distribution.