Final answer:
To find the minimal score for the bottom 22% of a normal distribution with a mean of 26 and a standard deviation of 4, locate the z-score corresponding to the 22nd percentile, approximately -0.77, and use the formula X = μ + zσ to calculate the score, roughly 22.92.
Step-by-step explanation:
To find the minimum score needed to be in the bottom 22% of a normal distribution, you would use the mean and standard deviation of the distribution along with a z-table or standard normal distribution table. Given the mean (μ) is 26 and the standard deviation (σ) is the square root of the variance, which is 4 (since 22=16), we can look for the z-score that corresponds to the bottom 22%. Once the z-score is found, use the formula X = μ + zσ to calculate the specific value.
First, locate the z-score on the z-table that corresponds to the bottom 22%, which is approximately -0.77. Then plug the values into the formula:
- X = μ + zσ
- X = 26 + (-0.77)(4)
- X ≈ 26 - 3.08
- X ≈ 22.92
Therefore, the minimum score needed to be in the bottom 22% of the distribution is approximately 22.92.