Question
Using the empirical rule, Mrs. G found that 95% of the population is aged between 14 and 42.
1.1 Calculate the mean and the standard deviation
1.2 Find the boundaries that will cover 99.7% of the population.
1.3 Find the boundaries that will cover 68% of the population.
1.4 Find the percentage of the population aged between 21 and 42
1.5 Find the percentage of the population aged more than 42
1.6 Find the percentage of the population aged 7 and 21
1.7 Find the percentage of the population aged less than 14
1.8 If a random sample of 525 is taken, how many people will be aged between 35 and 42
1.9 If a random sample of 1000 is taken, how many people will be aged more than 49.
1.1 Calculate the mean and the standard deviation
1.2 Find the boundaries that will cover 99.7% of the population.
1.3 Find the boundaries that will cover 68% of the population.
1.4 Find the percentage of the population aged between 21 and 42
1.5 Find the percentage of the population aged more than 42
1.6 Find the percentage of the population aged 7 and 21
1.7 Find the percentage of the population aged less than 14
1.8 If a random sample of 525 is taken, how many people will be aged between 35 and 42
1.9 If a random sample of 1000 is taken, how many people will be aged more than 49.
Answers
GPT-4o mini
To solve the questions, we first need to understand the empirical rule, also known as the 68-95-99.7 rule, which states that in a normal distribution:
- About 68% of the data falls within 1 standard deviation of the mean.
- About 95% of the data falls within 2 standard deviations of the mean.
- About 99.7% of the data falls within 3 standard deviations of the mean.
We will use this information to make calculations based on the information provided: 95% of the population is aged between 14 and 42.
### 1.1 Calculate the mean and the standard deviation
Given that 95% of the population is between 14 and 42, we can assume that these values are 2 standard deviations below and above the mean:
- Mean (\( \mu \)) is the midpoint between 14 and 42.
\[
\mu = \frac{14 + 42}{2} = \frac{56}{2} = 28
\]
- To find the standard deviation (\( \sigma \)), we need to find the distance from the mean to one of the endpoints (either 14 or 42) and divide by 2:
The distance from the mean to 14 and 42 is:
\[
42 - 28 = 14 \quad \text{and} \quad 28 - 14 = 14
\]
So:
\[
\sigma = \frac{42 - 14}{4} = \frac{28}{4} = 7
\]
Thus, the mean (\( \mu \)) is 28, and the standard deviation (\( \sigma \)) is 7.
### 1.2 Find the boundaries that will cover 99.7% of the population.
According to the empirical rule:
- 99.7% falls within 3 standard deviations from the mean.
The boundaries can be calculated as:
\[
\text{Lower boundary} = \mu - 3\sigma = 28 - 3(7) = 28 - 21 = 7
\]
\[
\text{Upper boundary} = \mu + 3\sigma = 28 + 3(7) = 28 + 21 = 49
\]
Therefore, the boundaries that cover 99.7% of the population are 7 and 49.
### 1.3 Find the boundaries that will cover 68% of the population.
According to the empirical rule:
- 68% falls within 1 standard deviation from the mean.
The boundaries can be calculated as:
\[
\text{Lower boundary} = \mu - \sigma = 28 - 7 = 21
\]
\[
\text{Upper boundary} = \mu + \sigma = 28 + 7 = 35
\]
Therefore, the boundaries that cover 68% of the population are 21 and 35.
### 1.4 Find the percentage of the population aged between 21 and 42.
Using the empirical rule and the calculated boundaries:
- The age range of 21 to 28 is one standard deviation below the mean.
- The age range of 28 to 42 encompasses the remaining range up to 42, which is more than 2 standard deviations.
From the empirical rule:
- 68% is between 21 and 35.
- The range from 35 to 42 covers the additional portion.
- To find the exact percentage from 21 to 42:
- The age 42 is 2 standard deviations above the mean, so that corresponds to 95%.
- Since 68% falls within 1 standard deviation, the remaining ages (which are between 35 and 42) can be calculated as:
\[
\frac{42-35}{42-14} = \frac{7}{28} = 0.25 \text{ of 95% is } 0.25 \times 95\% \approx 23.75\%
\]
Thus:
\[
\text{Percentage between 21 and 42} \approx 68\% + 23.75\% = 91.75\%
\]
### 1.5 Find the percentage of the population aged more than 42.
Given 95% of the population is between 14 and 42, this means:
\[
100\% - 95\% = 5\%
\]
Hence, the percentage of the population aged more than 42 is approximately **5%**.
### 1.6 Find the percentage of the population aged between 7 and 21.
The cumulative percentage between 7 and 21 includes everything below 21, up to three standard deviations below the mean:
- 7 is three standard deviations below the mean.
Therefore, according to the empirical rule:
- 99.7% falls between 7 and 49.
- Since 68% is between 21 and 35, we have 16% below 21 to consider for (7, 21).
So:
\[
\text{Percentage aged between 7 and 21} = \frac{68\%}{2} = 34\%
\]
Thus:
\[
\frac{100\%-97.5\%}{2} = 50\% - 34\% = 16\%
\]
Hence, the percentage of the population aged between 7 and 21 is approximately **16%**.
### 1.7 Find the percentage of the population aged less than 14.
Since the age 14 is included in the boundaries for 95%, it means:
\[
\text{Less than 14} = 2.5\%
\]
So, the percentage of the population aged less than 14 is approximately **2.5%**.
### 1.8 If a random sample of 525 is taken, how many people will be aged between 35 and 42.
From previous calculations, we estimated that approximately **23.75%** of the population is between 35 and 42.
Let's calculate:
\[
0.2375 \times 525 \approx 124.69 \approx 125
\]
Thus, in a sample of 525, approximately **125 people** will be aged between 35 and 42.
### 1.9 If a random sample of 1000 is taken, how many people will be aged more than 49.
As determined previously, approximately **0.3%** of the population is aged more than 49.
For a sample of 1000:
\[
0.05 \times 1000 = 5
\]
Thus, in a sample of 1000, approximately **5 people** will be aged more than 49.
If you have any further questions or need additional assistance, feel free to ask!
- About 68% of the data falls within 1 standard deviation of the mean.
- About 95% of the data falls within 2 standard deviations of the mean.
- About 99.7% of the data falls within 3 standard deviations of the mean.
We will use this information to make calculations based on the information provided: 95% of the population is aged between 14 and 42.
### 1.1 Calculate the mean and the standard deviation
Given that 95% of the population is between 14 and 42, we can assume that these values are 2 standard deviations below and above the mean:
- Mean (\( \mu \)) is the midpoint between 14 and 42.
\[
\mu = \frac{14 + 42}{2} = \frac{56}{2} = 28
\]
- To find the standard deviation (\( \sigma \)), we need to find the distance from the mean to one of the endpoints (either 14 or 42) and divide by 2:
The distance from the mean to 14 and 42 is:
\[
42 - 28 = 14 \quad \text{and} \quad 28 - 14 = 14
\]
So:
\[
\sigma = \frac{42 - 14}{4} = \frac{28}{4} = 7
\]
Thus, the mean (\( \mu \)) is 28, and the standard deviation (\( \sigma \)) is 7.
### 1.2 Find the boundaries that will cover 99.7% of the population.
According to the empirical rule:
- 99.7% falls within 3 standard deviations from the mean.
The boundaries can be calculated as:
\[
\text{Lower boundary} = \mu - 3\sigma = 28 - 3(7) = 28 - 21 = 7
\]
\[
\text{Upper boundary} = \mu + 3\sigma = 28 + 3(7) = 28 + 21 = 49
\]
Therefore, the boundaries that cover 99.7% of the population are 7 and 49.
### 1.3 Find the boundaries that will cover 68% of the population.
According to the empirical rule:
- 68% falls within 1 standard deviation from the mean.
The boundaries can be calculated as:
\[
\text{Lower boundary} = \mu - \sigma = 28 - 7 = 21
\]
\[
\text{Upper boundary} = \mu + \sigma = 28 + 7 = 35
\]
Therefore, the boundaries that cover 68% of the population are 21 and 35.
### 1.4 Find the percentage of the population aged between 21 and 42.
Using the empirical rule and the calculated boundaries:
- The age range of 21 to 28 is one standard deviation below the mean.
- The age range of 28 to 42 encompasses the remaining range up to 42, which is more than 2 standard deviations.
From the empirical rule:
- 68% is between 21 and 35.
- The range from 35 to 42 covers the additional portion.
- To find the exact percentage from 21 to 42:
- The age 42 is 2 standard deviations above the mean, so that corresponds to 95%.
- Since 68% falls within 1 standard deviation, the remaining ages (which are between 35 and 42) can be calculated as:
\[
\frac{42-35}{42-14} = \frac{7}{28} = 0.25 \text{ of 95% is } 0.25 \times 95\% \approx 23.75\%
\]
Thus:
\[
\text{Percentage between 21 and 42} \approx 68\% + 23.75\% = 91.75\%
\]
### 1.5 Find the percentage of the population aged more than 42.
Given 95% of the population is between 14 and 42, this means:
\[
100\% - 95\% = 5\%
\]
Hence, the percentage of the population aged more than 42 is approximately **5%**.
### 1.6 Find the percentage of the population aged between 7 and 21.
The cumulative percentage between 7 and 21 includes everything below 21, up to three standard deviations below the mean:
- 7 is three standard deviations below the mean.
Therefore, according to the empirical rule:
- 99.7% falls between 7 and 49.
- Since 68% is between 21 and 35, we have 16% below 21 to consider for (7, 21).
So:
\[
\text{Percentage aged between 7 and 21} = \frac{68\%}{2} = 34\%
\]
Thus:
\[
\frac{100\%-97.5\%}{2} = 50\% - 34\% = 16\%
\]
Hence, the percentage of the population aged between 7 and 21 is approximately **16%**.
### 1.7 Find the percentage of the population aged less than 14.
Since the age 14 is included in the boundaries for 95%, it means:
\[
\text{Less than 14} = 2.5\%
\]
So, the percentage of the population aged less than 14 is approximately **2.5%**.
### 1.8 If a random sample of 525 is taken, how many people will be aged between 35 and 42.
From previous calculations, we estimated that approximately **23.75%** of the population is between 35 and 42.
Let's calculate:
\[
0.2375 \times 525 \approx 124.69 \approx 125
\]
Thus, in a sample of 525, approximately **125 people** will be aged between 35 and 42.
### 1.9 If a random sample of 1000 is taken, how many people will be aged more than 49.
As determined previously, approximately **0.3%** of the population is aged more than 49.
For a sample of 1000:
\[
0.05 \times 1000 = 5
\]
Thus, in a sample of 1000, approximately **5 people** will be aged more than 49.
If you have any further questions or need additional assistance, feel free to ask!