Final answer:
The area under the standard normal distribution curve to the right of z = 1.97 is found by subtracting the Z-table value for z = 1.97, which represents the area to the left, from 1. For z = 1.97, the area to the right is 1 - 0.975 = 0.025.
Step-by-step explanation:
To find the area under the curve to the right of z = 1.97 in a standard normal distribution, you need to locate this value in the Z-table. Typically, Z-tables provide the area to the left of a z-score.
To find the area to the right, we subtract the table value from 1. For z = 1.97, the Z-table gives an area to the left of approximately 0.975.
Therefore, the area to the right is equal to 1 - 0.975 = 0.025. This method is consistent with the approach you’d take if you were trying to find critical values such as z0.01 or to determine the probability in other scenarios mentioned such as the area between two z-scores or the area under the normal curve for specific distributions like X~ N(54, 8).
To calculate another example, if the Z-table shows that the area to the left of z = 1.28 is approximately 0.9, we would conclude that the area to the right is 1 - 0.9 = 0.1. Remember that the standard normal distribution has a mean of 0 and standard deviation of 1, making it easy to compare different z-scores.