Final answer:
The 47th percentile for the distribution of the number of widgets produced is calculated by converting that percentile to a Z-score and then applying the formula X = μ + Zσ, where μ is the mean (85.46) and σ is the standard deviation (12.77).
Step-by-step explanation:
The question asks what the 47th percentile is for a normally distributed number of widgets produced by a machine with a mean of 85.46 and a standard deviation of 12.77. To find the 47th percentile, we first need to use the standard normal (Z) distribution since the distribution of widgets follows a normal pattern. Percentiles in a normal distribution can be found using a Z-table, statistical software, or a calculator capable of providing normal distribution probabilities. The Z-score corresponding to the 47th percentile can be found, and then we use the formula:
X = μ + Zσ
Where:
- X is the value that corresponds to the percentile we want to find (47th percentile in this case)
- μ (mu) is the mean of the distribution
- Z is the Z-score associated with the 47th percentile from the standard normal distribution
- σ (sigma) is the standard deviation of the distribution
After obtaining the Z-score for the 47th percentile, you plug in the values into the formula to find the specific number of widgets that represents the 47th percentile.