Final answer:
To find the constant c for which P(|Z| ≤ c) = 0.762 in a standard normal distribution, one would use a z-table or statistical calculator to find the z-score corresponding to a cumulative area of 0.381 (half of 0.762) from the left. This z-score is the value of c.
Step-by-step explanation:
The question asks to find the value of the constant c such that P(|Z| ≤ c) = 0.762, where Z is a standard normal variable (Z ≈ N(0, 1)). To solve this, we typically use a z-table or a statistical calculator. The probability given is the total area under the standard normal curve between -c and c. Since the standard normal distribution is symmetric, half of the area will be to the left of 0, and half to the right. For P(|Z| ≤ c), we want the cumulative probability from -∞ to c, which is 0.381 (half of 0.762), to capture the left half, and then the table or calculator reflects that to the right side automatically.
Using a statistical calculator like the TI-83, 83+, or 84+, we would enter the command invNorm(0.881,0,1) to find Z0.025, knowing that the area to the right is 0.025 and the area to the left is 0.975. Alternatively, by using a z-table or online tool, you would look up the z-score that corresponds to a cumulative area of 0.381. You would find that the z-score closest to where the cumulative area from the left is 0.381 in the standard normal distribution, and this z-score is your value of c.