Sure, let's walk through the process of finding the z-score that corresponds to the 65th percentile in a standard normal distribution.
1. Understanding Percentiles in a Standard Normal Distribution:
- A standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1.
- The percentile represents the point below which a given percentage of observations fall. So, a 65th percentile means that 65% of the data falls below this point.
2. Using Standard Normal Distribution Tables (z-tables):
- To find the z-score for a given percentile, we can refer to standard normal distribution tables which provide the area (or cumulative probability) to the left of a given z-score.
- The 65th percentile corresponds to an area of 0.65 to the left of the desired z-score.
3. Locating the 65th Percentile in the Z-Table:
- If we look up the cumulative probability of 0.65 in the z-table or use statistical software or calculators designed for this purpose, we would find the z-score that matches this probability.
From the above steps, the z-score that corresponds to the 65th percentile in a standard normal distribution is approximately:
[tex]\[ \boxed{0.3853} \][/tex]
So, the z-score corresponding to the 65th percentile in a standard normal distribution is approximately 0.3853.
