To find the probability that a given tree in the orchard has between 240 and 330 apples, we can utilize the principles of the normal distribution and standard deviation.
1. Identify the Mean and Standard Deviation:
- The mean number of apples ([tex]\(\mu\)[/tex]) is 300.
- The standard deviation ([tex]\(\sigma\)[/tex]) is 30.
2. Determine the Bounds:
- The lower bound ([tex]\(X_{lower}\)[/tex]) is 240 apples.
- The upper bound ([tex]\(X_{upper}\)[/tex]) is 330 apples.
3. Calculate the Z-scores:
- The Z-score measures how many standard deviations an element is from the mean.
- For the lower bound ([tex]\(Z_{lower}\)[/tex]):
[tex]\[
Z_{lower} = \frac{X_{lower} - \mu}{\sigma} = \frac{240 - 300}{30} = -2
\][/tex]
- For the upper bound ([tex]\(Z_{upper}\)[/tex]):
[tex]\[
Z_{upper} = \frac{X_{upper} - \mu}{\sigma} = \frac{330 - 300}{30} = 1
\][/tex]
4. Interpret the Z-scores:
- A Z-score of -2 means 240 is 2 standard deviations below the mean.
- A Z-score of 1 means 330 is 1 standard deviation above the mean.
5. Find the Corresponding Probabilities Using the Standard Normal Distribution:
- Using standard normal distribution tables or cumulative distribution function (CDF):
- The cumulative probability up to [tex]\(Z_{lower} = -2\)[/tex] is approximately 0.0228.
- The cumulative probability up to [tex]\(Z_{upper} = 1\)[/tex] is approximately 0.8413.
6. Calculate the Desired Probability:
- The probability that a number of apples falls between 240 and 330 apples is the difference between the cumulative probabilities at [tex]\(Z_{upper}\)[/tex] and [tex]\(Z_{lower}\)[/tex]:
[tex]\[
P(240 < a < 330) = P(Z_{upper}) - P(Z_{lower}) = 0.8413 - 0.0228 = 0.8185
\][/tex]
Therefore, the probability that a given tree has between 240 and 330 apples is approximately [tex]\(0.8185\)[/tex] or [tex]\(81.85\%\)[/tex].
