Answer:81.86
Step-by-step explanation:
The probability that a given tree has between 240 and 330 apples can be calculated using the 68%-95%-99.7% rule for a normally distributed variable.
Given:
Mean (μ) = 300 apples
Standard deviation (σ) = 30 apples
We need to find the probability that the number of apples falls within the range of 240 to 330. Let’s calculate it step by step:
Calculate the z-scores for 240 and 330:
(z_{240} = \frac{{240 - \mu}}{{\sigma}})
(z_{330} = \frac{{330 - \mu}}{{\sigma}})
Use the cumulative distribution function (CDF) of the standard normal distribution to find the probabilities:
(P(240 < a < 330) = P(z_{330}) - P(z_{240}))
Convert the result to a percentage.
Let’s compute it:
(z_{240} = \frac{{240 - 300}}{{30}} = -2)
(z_{330} = \frac{{330 - 300}}{{30}} = 1)
Using the standard normal distribution table or a calculator, we find:
(P(z_{240}) \approx 0.0228)
(P(z_{330}) \approx 0.8466)
Therefore:
(P(240 < a < 330) \approx 0.8466 - 0.0228 = 0.8238)
Converting to a percentage:
(P(240 < a < 330) \approx 81.86%)
So, the probability that a given tree has between 240 and 330 apples is approximately 81.86%.
