Final answer:
To find the probability that Z^2 is less than 1, we need to calculate the area under the chi-squared distribution curve with 1 degree of freedom that corresponds to values less than 1.
Step-by-step explanation:
To find P(Z^2 < 1), we need to find the area under the standard normal distribution curve that corresponds to values of Z^2 less than 1. Since Z is a standard normal random variable, Z^2 will follow a chi-squared distribution with 1 degree of freedom. The chi-squared distribution is right-skewed and takes only positive values.
The probability that Z^2 is less than 1 is equivalent to the probability that chi-squared with 1 degree of freedom is less than 1. We can calculate this probability using a chi-squared distribution table or a calculator.
Using a calculator, we can find that P(Z^2 < 1) is approximately 0.6827. This means that there is about a 68.27% chance that Z^2 is less than 1.