Questions: Salaries for teachers in a particular elementary school district are normally distributed with a mean of 66,400 and a standard deviation of 1,800. We randomly survey 8 teachers from that district. Find the mean and standard deviation of the distribution of sums; round to three decimal places when needed. μ∑x= σ∑x= Find the 85th percentile for the sum of the salaries of 8 teachers. Round to the nearest dollar. x=

Salaries for teachers in a particular elementary school district are normally distributed with a mean of 66,400 and a standard deviation of 1,800. We randomly survey 8 teachers from that district. Find the mean and standard deviation of the distribution of sums; round to three decimal places when needed. μ∑x= σ∑x= Find the 85th percentile for the sum of the salaries of 8 teachers. Round to the nearest dollar. x=
Transcript text: Salaries for teachers in a particular elementary school district are normally distributed with a mean of $\$ 66,400$ and a standard deviation of $\$ 1,800$. We randomly survey 8 teachers from that district. Find the mean and standard deviation of the distribution of sums; round to three decimal places when needed. \[ \begin{array}{l} \mu_{\sum\{x\}}=\square \\ \sigma_{\sum\{x\}}=\square \end{array} \] Find the 85 th percentile for the sum of the salaries of 8 teachers. Round to the nearest dollar. \[ x= \] $\square$
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Solution

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Solution Steps

Step 1: Mean of the Distribution of Sums

The mean of the distribution of sums for the salaries of 8 teachers is calculated as follows:

\[ \mu_{\sum\{x\}} = n \cdot \mu = 8 \cdot 66400 = 531200 \]

Step 2: Standard Deviation of the Distribution of Sums

The standard deviation of the distribution of sums is determined using the formula:

\[ \sigma_{\sum\{x\}} = \sigma \cdot \sqrt{n} = 1800 \cdot \sqrt{8} \approx 5091.169 \]

Step 3: Z-Score for the 85th Percentile

To find the 85th percentile, we first calculate the z-score corresponding to this percentile. The z-score for the 85th percentile is:

\[ z = 0.85 \]

Step 4: 85th Percentile for the Sum of Salaries

Using the z-score, we can find the 85th percentile for the sum of the salaries of 8 teachers:

\[ X = \mu_{\sum\{x\}} + z \cdot \sigma_{\sum\{x\}} = 531200 + 0.85 \cdot 5091.169 \approx 535527 \]

Final Answer

The results are as follows: \[ \mu_{\sum\{x\}} = 531200, \quad \sigma_{\sum\{x\}} \approx 5091.169, \quad \text{85th percentile} \approx 535527 \]

Thus, the final boxed answers are: \[ \boxed{\mu_{\sum\{x\}} = 531200} \] \[ \boxed{\sigma_{\sum\{x\}} \approx 5091.169} \] \[ \boxed{X \approx 535527} \]

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