Questions: Assume that the salaries of elementary school teachers in the United States are normally distributed with a mean of 56,800 and a standard deviation of 2,300. A. What is the cutoff salary for teachers in the top 10% ? Round to 2 decimals. B. What is the salary amounts for the middle 60% of elementary school teachers based on this information? Round to 2 decimals. Between and

Solution Steps

To solve these problems, we will use properties of the normal distribution.

A. To find the cutoff salary for the top 10%, we need to determine the salary corresponding to the 90th percentile of the distribution. This can be done using the inverse cumulative distribution function (CDF) for a normal distribution.

B. To find the salary range for the middle 60%, we need to determine the 20th and 80th percentiles of the distribution. These percentiles will give us the lower and upper bounds of the salary range.

To find the cutoff salary for the top 10% of elementary school teachers, we need to calculate the 90th percentile of a normal distribution with a mean (\(\mu\)) of \$56,800 and a standard deviation (\(\sigma\)) of \$2,300. The 90th percentile corresponds to the point where 90% of the data falls below it.

Using the inverse cumulative distribution function, we find:

\[ \text{Cutoff Salary} = \mu + \sigma \times z_{0.90} \]

where \(z_{0.90}\) is the z-score for the 90th percentile. The calculated cutoff salary is approximately \$59,747.57.

To find the salary range for the middle 60% of teachers, we need to calculate the 20th and 80th percentiles of the distribution. These percentiles will give us the lower and upper bounds of the salary range.

The 20th percentile corresponds to the point where 20% of the data falls below it, and the 80th percentile corresponds to the point where 80% of the data falls below it.

Using the inverse cumulative distribution function, we find:

\[ \text{Lower Salary} = \mu + \sigma \times z_{0.20} \] \[ \text{Upper Salary} = \mu + \sigma \times z_{0.80} \]

The calculated lower salary is approximately \$54,864.27, and the upper salary is approximately \$58,735.73.

Final Answer