A paint machine dispenses dye into paint cans to create various shades. The amount of dye dispensed per can follows a normal distribution with a mean of 5 milliliters (ml) and a standard deviation of 0.4 ml. Determine the amount of dye corresponding to the 9th percentile of this distribution.

To find the amount of dye corresponding to the 9th percentile of the distribution, we need to use the properties of the normal distribution. The 9th percentile (P9) indicates that 9% of the values in the distribution fall below this amount.

To find this, we can use a standard normal distribution table (Z-table) or a statistical calculator that allows input for percentile values to find the Z-score corresponding to the 9th percentile.

Here are the steps:

1. Locate the Z-score for the 9th percentile using the Z-table or a calculator. 2. Since the Z-score represents the number of standard deviations a value is from the mean, use the following formula to find the actual value corresponding to this Z-score: \( X = \mu + Z \times \sigma \) Where: - X is the value we want to find (the 9th percentile). - μ is the mean of the distribution (5 ml). - Z is the Z-score for the 9th percentile. - σ is the standard deviation of the distribution (0.4 ml).

For example, let's say the Z-score for the 9th percentile is approximately -1.34. We can calculate X as follows:

\( X = 5 + (-1.34) \times 0.4 \) \( X = 5 - 0.536 \) \( X = 4.464 \)

Therefore, the amount of dye corresponding to the 9th percentile of this distribution is approximately 4.464 ml.