The mean value of land and buildings per acre from a sample of farms is

​$13001300​,

with a standard deviation of

​$100100.

The data set has a​ bell-shaped distribution. Assume the number of farms in the sample is

7474.

Question content area bottom

Part 1

​(a)**** Use the empirical rule to estimate the number of farms whose land and building values per acre are between

​$12001200

and

​$14001400.

5050

farms ​(Round to the nearest whole number as​ needed.)****

Part 2

​(b) If

2121

additional farms were​ sampled, about how many of these additional farms would you expect to have land and building values between

​$12001200

per acre and

​$14001400

per​ acre?

enter your response here

farms out of

2121

​(Round to the nearest whole number as​ needed.)

Let's solve these problems step-by-step using the empirical rule (also known as the 68-95-99.7 rule) for normally distributed data. 

### Part 1: Estimating the Number of Farms Within $1200 and $1400

#### Given:
- Mean ($\mu$) = $1300
- Standard deviation ($\sigma$) = $100
- Sample size = 74 farms
- Interval: $1200 to $1400

According to the empirical rule for a normal distribution:
- Approximately **68%** of the data falls within 1 standard deviation ($\sigma$) of the mean.
- In this case, 1 standard deviation from the mean is:
  \[
  \text{Lower bound} = \mu - \sigma = 1300 - 100 = 1200
  \]
  \[
  \text{Upper bound} = \mu + \sigma = 1300 + 100 = 1400
  \]

So, approximately **68%** of the farms have land and building values per acre between $1200 and $1400.

To find the number of farms, multiply 68% by the total number of farms:

\[
\text{Number of farms} = 0.68 \times 74 \approx 50
\]

Thus, the estimated number of farms whose land and building values are between $1200 and $1400 is **50 farms**.

### Part 2: Estimating the Number of Additional Farms Within $1200 and $1400

If **21 additional farms** are sampled, we want to estimate how many of these farms would have values between $1200 and $1400 per acre.

Since the percentage of farms falling within 1 standard deviation of the mean is still **68%**, we use the same percentage to estimate the number of additional farms:

\[
\text{Number of additional farms} = 0.68 \times 21 \approx 14
\]

So, about **14 out of 21 additional farms** would be expected to have land and building values between $1200 and $1400 per acre.

### Summary
- **Part 1:** Estimated number of farms within $1200 and $1400: **50 farms**.
- **Part 2:** Estimated number of additional farms within $1200 and $1400: **14 farms**.

---

Would you like more details on any of these steps, or do you have any other questions? 

Here are 5 related questions to expand your understanding:
1. How does the empirical rule apply to data that is not normally distributed?
2. What is the significance of standard deviation in data analysis?
3. How would the results change if the distribution were not bell-shaped?
4. How can you calculate the percentage of data within two standard deviations of the mean?
5. What are other methods for determining the spread of data in a dataset?

**Tip:** Remember that the empirical rule is most accurate when the dataset follows a normal (bell-shaped) distribution.

The mean value of land and buildings per acre from a sample of farms is $1300, with a standard deviation of $100. The data set has a bell-shaped distribution. Assume the number of farms in the sample is 74. Part 1 (a) Use the empirical rule to estimate the number of farms whose land and building values per acre are between $1200 and $1400. 50 farms (Round to the nearest whole number as needed.) Part 2 (b) If 21 additional farms were sampled, about how many of these additional farms would you expect to have land and building values between $1200 per acre and $1400 per acre? farms out of 21 (Round to the nearest whole number as needed.)

Solve a statistics problem using the empirical rule to estimate the number of farms with land values between $1200 and $1400 per acre. The data follows a bell-shaped distribution with a mean of $1300 and a standard deviation of $100. Learn how to apply the empirical rule to a sample of 74 farms and 21 additional farms.

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