Answer :
Certainly! Let's tackle this problem step-by-step by following the 68%-95%-99.7% rule, which is also known as the empirical rule in statistics. This rule provides insight into how a dataset's values are distributed around its mean, assuming the dataset follows a normal distribution.
### Step-by-Step Solution:
1. Calculate the Mean:
The mean (average) number of apples can be calculated from the given lower and upper bounds:
[tex]\[ \text{Mean} = \frac{330 + 390}{2} = 360 \][/tex]
2. Calculate the Range and Standard Deviation:
The total range of apples is given as the difference between the upper and lower bounds:
[tex]\[ \text{Range} = 390 - 330 = 60 \][/tex]
The 95% confidence interval in a normal distribution is roughly equivalent to \( \pm 2\) standard deviations from the mean. Therefore, to find the standard deviation (\(\sigma\)):
[tex]\[ 60 \approx 4\sigma \quad \Rightarrow \quad \sigma = \frac{60}{4} = 15 \][/tex]
3. Calculate the 68% Interval:
The 68% interval (one standard deviation) means we expect 68% of the values to fall within \( \pm 1 \sigma\):
[tex]\[ \text{68% interval} = [\text{Mean} - \sigma, \text{Mean} + \sigma] = [360 - 15, 360 + 15] = [345, 375] \][/tex]
4. Calculate the 95% Interval:
The 95% interval (two standard deviations) means we expect 95% of the values to fall within \( \pm 2 \sigma\):
[tex]\[ \text{95% interval} = [\text{Mean} - 2\sigma, \text{Mean} + 2\sigma] = [360 - 30, 360 + 30] = [330, 390] \][/tex]
5. Calculate the 99.7% Interval:
The 99.7% interval (three standard deviations) means we expect 99.7% of the values to fall within \( \pm 3 \sigma\):
[tex]\[ \text{99.7% interval} = [\text{Mean} - 3\sigma, \text{Mean} + 3\sigma] = [360 - 45, 360 + 45] = [315, 405] \][/tex]
### Summary of Results:
- Mean number of apples: \(360.0\)
- Standard deviation: \(15.0\)
- 68% interval: \([345.0, 375.0]\)
- 95% interval: \([330.0, 390.0]\)
- 99.7% interval: \([315.0, 405.0]\)
These results provide a detailed understanding of the distribution of the apple counts on the given tree. The intervals show where the majority of the data will likely fall given the normal distribution assumption.
### Step-by-Step Solution:
1. Calculate the Mean:
The mean (average) number of apples can be calculated from the given lower and upper bounds:
[tex]\[ \text{Mean} = \frac{330 + 390}{2} = 360 \][/tex]
2. Calculate the Range and Standard Deviation:
The total range of apples is given as the difference between the upper and lower bounds:
[tex]\[ \text{Range} = 390 - 330 = 60 \][/tex]
The 95% confidence interval in a normal distribution is roughly equivalent to \( \pm 2\) standard deviations from the mean. Therefore, to find the standard deviation (\(\sigma\)):
[tex]\[ 60 \approx 4\sigma \quad \Rightarrow \quad \sigma = \frac{60}{4} = 15 \][/tex]
3. Calculate the 68% Interval:
The 68% interval (one standard deviation) means we expect 68% of the values to fall within \( \pm 1 \sigma\):
[tex]\[ \text{68% interval} = [\text{Mean} - \sigma, \text{Mean} + \sigma] = [360 - 15, 360 + 15] = [345, 375] \][/tex]
4. Calculate the 95% Interval:
The 95% interval (two standard deviations) means we expect 95% of the values to fall within \( \pm 2 \sigma\):
[tex]\[ \text{95% interval} = [\text{Mean} - 2\sigma, \text{Mean} + 2\sigma] = [360 - 30, 360 + 30] = [330, 390] \][/tex]
5. Calculate the 99.7% Interval:
The 99.7% interval (three standard deviations) means we expect 99.7% of the values to fall within \( \pm 3 \sigma\):
[tex]\[ \text{99.7% interval} = [\text{Mean} - 3\sigma, \text{Mean} + 3\sigma] = [360 - 45, 360 + 45] = [315, 405] \][/tex]
### Summary of Results:
- Mean number of apples: \(360.0\)
- Standard deviation: \(15.0\)
- 68% interval: \([345.0, 375.0]\)
- 95% interval: \([330.0, 390.0]\)
- 99.7% interval: \([315.0, 405.0]\)
These results provide a detailed understanding of the distribution of the apple counts on the given tree. The intervals show where the majority of the data will likely fall given the normal distribution assumption.
