To solve these problems, we can use the concept of the standard normal distribution, also known as the Z-score.
For the first problem:
To find the cutoff salary for teachers in the bottom 10%, we need to find the Z-score associated with the 10th percentile and then convert it back to an actual salary value.
1. Calculate the Z-score:
The Z-score formula is (X - μ) / σ, where X is the value (salary) you want to convert, μ is the mean of the distribution, and σ is the standard deviation.
Z = (X - μ) / σ
2. Look up the Z-score:
Use a Z-table or calculator to find the Z-score corresponding to the 10th percentile. The Z-score represents the number of standard deviations below the mean.
3. Convert back to salary:
Multiply the Z-score by the standard deviation and add it to the mean to find the cutoff salary for the bottom 10%.
For the second problem:
To find the time that boys need to beat in order to earn a certificate of recognition, we follow a similar process.
1. Calculate the Z-score using the formula Z = (X - μ) / σ, where X is the time you want to convert, μ is the mean, and σ is the standard deviation.
2. Look up the Z-score corresponding to the top 10% in a Z-table or calculator. The Z-score represents the number of standard deviations above the mean.
3. Convert back to time by multiplying the Z-score by the standard deviation and adding it to the mean. This will give you the time the boys need to beat to earn a certificate of recognition.
Please provide the specific values (X, μ, and σ) for each problem so I can guide you through the calculations.