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.) \( \$ 1774, \$ 1536, \$ 1341, \$ 428, \$ 601, \$ 1062, \$ 947, \$ 796 \)

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: \( \mu - 2\sigma = 1,100 - (2 \times 210) = 680 \) - Upper Bound: \( \mu + 2\sigma = 1,100 + (2 \times 210) = 1,520 \) 2. **3 Standard Deviations from the Mean**: - Lower Bound: \( \mu - 3\sigma = 1,100 - (3 \times 210) = 470 \) - Upper Bound: \( \mu + 3\sigma = 1,100 + (3 \times 210) = 1,730 \) ### 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.