To solve this problem, we will use the properties of the normal distribution and the empirical rule, also known as the 68%-95%-99.7% rule.
First, let's outline the information given:
- Mean number of apples (\(\mu\)): 300 apples
- Standard deviation (\(\sigma\)): 30 apples
We need to find the probability that the number of apples on a given tree is between 240 and 300.
### Step-by-Step Solution:
1. Calculate the Z-scores for the bounds (240 and 300):
The Z-score is a measure of how many standard deviations an element is from the mean. It is calculated using the formula:
[tex]\[
Z = \frac{{X - \mu}}{\sigma}
\][/tex]
For the lower bound (240 apples):
[tex]\[
Z_{\text{lower}} = \frac{240 - 300}{30} = \frac{-60}{30} = -2
\][/tex]
For the upper bound (300 apples):
[tex]\[
Z_{\text{upper}} = \frac{300 - 300}{30} = \frac{0}{30} = 0
\][/tex]
2. Interpret the Z-scores using the empirical rule:
According to the empirical rule, approximately:
- 68% of the data lies within 1 standard deviation of the mean.
- 95% of the data lies within 2 standard deviations of the mean.
- 99.7% of the data lies within 3 standard deviations of the mean.
Here, a Z-score of -2 corresponds to being 2 standard deviations below the mean, while a Z-score of 0 corresponds to the mean itself.
3. Determine the cumulative probabilities:
Using the empirical rule:
- The cumulative probability of being within 2 standard deviations below the mean and 2 standard deviations above the mean (from Z = -2 to Z = 2) is approximately 95%. Since this interval is symmetric, it includes 47.5% of the data below the mean and 47.5% above the mean.
- Since 240 apples is 2 standard deviations below the mean, the probability of having more than 240 apples is approximately \(50\% + 47.5\% = 97.5\%\).
4. Find the probability between the bounds:
Since we need the probability between 240 and 300 apples:
- The cumulative probability up to the mean (300 apples) is \(50\%\).
- The cumulative probability up to 240 apples (2 standard deviations below the mean) is given by the empirical rule as \(2.5\%\) (because \(100\% - 97.5\%\)).
Therefore, the probability of a tree having between 240 and 300 apples is the difference between these cumulative probabilities.
[tex]\[
P(240 < a < 300) = 50\% - 2.5\% = 0.50 - 0.025 = 0.475
\][/tex]
Finally, to express this as a percentage:
[tex]\[
P(240 < a < 300) = 47.5\%
\][/tex]
This calculation aligns with the detailed step-by-step process provided by the values obtained.
