The mass, in grams of certain species of insects is uniformly distributed on the interval [2,10]. What proportion of insects are between 5.61g and 7.20g? Round final answer to 2 decimal places

To solve this problem, we'll use the concept of uniform distribution. For a uniform distribution on the interval \([a, b]\), the probability that a random variable \(X\) falls within an interval \([c, d]\) is given by:

\[
P(c \leq X \leq d) = \frac{d - c}{b - a}
\]

Given:
- \(a = 2\) grams (lower bound of the distribution)
- \(b = 10\) grams (upper bound of the distribution)
- \(c = 5.61\) grams
- \(d = 7.20\) grams

Now, substitute the values into the formula:

\[
P(5.61 \leq X \leq 7.20) = \frac{7.20 - 5.61}{10 - 2}
\]

Let's calculate this.

\[
P(5.61 \leq X \leq 7.20) = \frac{1.59}{8} \approx 0.19875
\]

Rounding to two decimal places:

\[
P(5.61 \leq X \leq 7.20) \approx 0.20
\]

So, the proportion of insects that are between 5.61g and 7.20g is approximately **0.20** or 20%.

Would you like to go into more detail about this calculation or the concept of uniform distribution? Here are some related questions to further explore:

1. What is the probability of an insect being less than 4 grams?
2. How would you calculate the mean mass of the insects?
3. What is the variance of the mass distribution for these insects?
4. How would the proportion change if the interval was [4, 8] grams?
5. How does the uniform distribution differ from a normal distribution?

**Tip:** In a uniform distribution, each interval of the same length has the same probability, making it a useful model for situations where outcomes are equally likely within a certain range.

Understand how to calculate the proportion of insects between 5.61 grams and 7.20 grams in a uniform distribution with a detailed step-by-step solution. This problem involves applying the concept of uniform distribution and calculating probabilities within a specified interval.

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