Question

The mean value of land and buildings per acre from a sample of farms is 1200, with a standard deviation of 100. The data set has a bell-shaped distribution. Using the empirical rule, determine which of the following farms, whose land and building values per acre are given, are unusual (more than two standard deviations from the mean). Are any of the data values very unusual (more than three standard deviations from the mean)?

1) 1034
2) 1445
3) 1043
4) 844
5) 1280
6) 1348

Answer 1

Farm values of 844 and 1445 are unusual, while 844 is very unusual.

Step-by-step explanation:

The empirical rule states that within a bell-shaped distribution:

- Approximately 68% of the data lies within one standard deviation from the mean.

- Around 95% of the data lies within two standard deviations.

- Almost all data (about 99.7%) lies within three standard deviations.

Given the mean of 1200 and a standard deviation of 100, two standard deviations from the mean would be \(1200 \pm (2 \times 100) = 1000\) to 1400.

1. **1034**: Within two standard deviations (1200 ± 2SD), falls within the range (1000-1400). Not unusual.

2. **1445**: Beyond two standard deviations, it's considered unusual (more than 2SD above the mean).

3. **1043**: Falls within two standard deviations (1000-1400). Not unusual.

4. **844**: Beyond two standard deviations, considered unusual (more than 2SD below the mean).

5. **1280**: Within two standard deviations (1000-1400). Not unusual.

6. **1348**: Within two standard deviations (1000-1400). Not unusual.

Next, considering three standard deviations from the mean (1200 ± 3SD = 900 to 1500), any value outside this range would be very unusual.

- 844 is more than 3SD below the mean, making it very unusual.

In summary, farms with values of 844 and 1445 are unusual, with 844 being very unusual as it lies more than three standard deviations below the mean, according to the empirical rule applied to the bell-shaped distribution of the dataset.