25. The mean monthly rent for a sample of studio apartments in one city is $$ \$ 1100 $$ with a standard deviation of $$ \$ 210 $$. The monthly rents for eight more studio apartments in the city a listed. Using the sample statistics above, determine which of the data values are unusual. Are any of the data values very unusual? Explain. (Assume the data set has a bell-shaped distribution.) $$ \$ 1774, \$ 1536, \$ 1341, \$ 428, \$ 601, \$ 1062, \$ 947, \$ 796 $$

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To determine which data values are **unusual** or **very unusual** in the given dataset, we'll use the concepts of **standard deviations** in a **bell-shaped (normal) distribution**.

### Given:
- **Mean ($$ \mu $$)**: \$1,100
- **Standard Deviation ($$ \sigma $$)**: \$210
- **Data Values**: \$1,774, \$1,536, \$1,341, \$428, \$601, \$1,062, \$947, \$796

### Definitions:
1. **Unusual Values**: Data points that are more than **2 standard deviations** away from the mean.
2. **Very Unusual Values**: Data points that are more than **3 standard deviations** away from the mean.

### Calculations:

1. **2 Standard Deviations from the Mean**:
   - Lower Bound: $$ \mu - 2\sigma = 1,100 - (2 	imes 210) = 680 $$
   - Upper Bound: $$ \mu + 2\sigma = 1,100 + (2 	imes 210) = 1,520 $$

2. **3 Standard Deviations from the Mean**:
   - Lower Bound: $$ \mu - 3\sigma = 1,100 - (3 	imes 210) = 470 $$
   - Upper Bound: $$ \mu + 3\sigma = 1,100 + (3 	imes 210) = 1,730 $$

### Analyzing Each Data Value:

| **Value (\$)** | **Position Relative to Bounds**                | **Classification**  |
|----------------|------------------------------------------------|---------------------|
| **1,774**      | **Above** $$ 1,730 $$ (Mean + 3σ)              | **Very Unusual**    |
| **1,536**      | Between $$ 1,520 $$ and $$ 1,730 $$ (2σ - 3σ)   | **Unusual**         |
| **1,341**      | Between $$ 680 $$ and $$ 1,520 $$ (-2σ to +2σ)   | **Not Unusual**     |
| **428**        | **Below** $$ 470 $$ (Mean - 3σ)                | **Very Unusual**    |
| **601**        | Between $$ 470 $$ and $$ 680 $$ (-3σ to -2σ)    | **Unusual**         |
| **1,062**      | Between $$ 680 $$ and $$ 1,520 $$ (-2σ to +2σ)   | **Not Unusual**     |
| **947**        | Between $$ 680 $$ and $$ 1,520 $$ (-2σ to +2σ)   | **Not Unusual**     |
| **796**        | Between $$ 680 $$ and $$ 1,520 $$ (-2σ to +2σ)   | **Not Unusual**     |

### Summary:

- **Very Unusual Values**:
  - **\$1,774**: Exceeds the upper limit by more than 3 standard deviations.
  - **\$428**: Falls below the lower limit by more than 3 standard deviations.

- **Unusual Values**:
  - **\$1,536**: Exceeds the upper limit by between 2 and 3 standard deviations.
  - **\$601**: Falls below the lower limit by between 2 and 3 standard deviations.

- **Not Unusual**:
  - **\$1,341**, **\$1,062**, **\$947**, **\$796**: All lie within 2 standard deviations of the mean.

### Conclusion:

- **Unusual Data Values**: \$1,536 and \$601.
- **Very Unusual Data Values**: \$1,774 and \$428.

These classifications are based on their positions relative to the mean and standard deviations in a normally distributed dataset.

**Answer Summary:**
All rents except $1,774 and $428 fall within two standard deviations of the mean and are not unusual. The values $1,536 and $601 are more than two standard deviations from the mean and are therefore unusual. Additionally, $1,774 and $428 lie beyond three standard deviations and are very unusual.
The unusual rents are \$1,536, \$601, \$1,774, and \$428.

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