Can someone assit me with my two math problems
Assume that the salaries of elementary school teachers in the united states are normally distributed with the mean of 36,000 and a standard deviation of 6,000. What is the cut off salary for teachers in the bottom 10%?
A physical fitness association is including the mile run in its secondary-school fitness test. The time for this event for boys in secondary school is known to possess a normal distribution with a mean of 450 seconds and a standard deviation of 40 seconds. The fitness association wants to recognize the fastest 10% of the boys with certificates of recognition. What time would the boys need to beat in order to earn a certificate of recognition from the fitness association?
Z = (raw score - mean)/standard deviation
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion in smallest area = .10. Insert that Z score and your other data in the equation above to get your answer.
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.