Final answer:
To find the 40th percentile (P₄₀) in a normal distribution with a mean of 23.3 and a standard deviation of 79.6, one would use the z-score for the 40th percentile and transform it into the actual score using the formula Score = mean + (z-score × standard deviation).
Step-by-step explanation:
To find P₄₀, which is the score separating the bottom 40% from the top 60% in a normal distribution with a mean of 23.3 and a standard deviation of 79.6, one would typically use a standard normal distribution table or a calculator with the inverse cumulative distribution function for the normal distribution. This value is also known as the 40th percentile. The 40th percentile corresponds to the z-score that has 40% of the distribution's area to its left. To find this z-score, you can use statistical software, a calculator, or a z-table. Once the z-score is found, it is then converted to the actual score using the mean and standard deviation of the distribution:
Score = mean + (z-score × standard deviation).
For example, if the z-score corresponding to the 40th percentile is -0.25 (this is just illustrative and not the actual value for P₄₀), the score separating the bottom 40% would be calculated as follows:
Score = 23.3 + (-0.25 × 79.6) = 23.3 - 19.9 = 3.4.