Normal VariablesWhat is the probability of a normally distributed random variable lying within 1.65 standard deviations of the mean?

The probability of a normally distributed random variable lying within 1.65 standard deviations of the mean is approximately 90%.

Here's the step-by-step explanation:

1. In a normal distribution, the mean, median and mode are all the same and are located at the center of the distribution.

2. The standard deviation is a measure of the amount of variation or dispersion in the set of values.

3. In a normal distribution, about 68% of the data falls within one standard deviation of the mean, about 95% falls within two standard deviations, and about 99.7% falls within three standard deviations.

4. Therefore, if we want to find the probability of a random variable lying within 1.65 standard deviations of the mean, we can use the empirical rule (also known as the 68-95-99.7 rule).

5. The empirical rule doesn't give us an exact percentage for 1.65 standard deviations, but we know it's somewhere between 68% (for 1 standard deviation) and 95% (for 2 standard deviations).

6. Looking at a standard normal distribution table, or using a calculator or software that can calculate probabilities for a normal distribution, we find that the probability is approximately 90%.

So, there's about a 90% chance that a normally distributed random variable will fall within 1.65 standard deviations of the mean.