To solve the problem regarding the Binomial random variable $X$ with parameters $n = 59$ and $p = 0.49$, we can proceed as follows:

Part I: Compute the Mean, Variance, and Standard Deviation

1. Mean ($\mu$): $\mu = n \times p = 59 \times 0.49 = 28.91$

2. Variance ($\sigma^2$): $\sigma^2 = n \times p \times (1 - p) = 59 \times 0.49 \times (1 - 0.49) = 59 \times 0.49 \times 0.51$ $\sigma^2 = 59 \times 0.49 \times 0.51 \approx 14.7861$

3. Standard Deviation ($\sigma$): $\sigma = \sqrt{\sigma^2} \approx \sqrt{14.7861} \approx 3.8431$

Part II: Probability Values

For the probability calculations, we can use a binomial calculator or statistical software to find:

4. Probability $P(X > 35)$: Using a binomial distribution calculator: $P(X > 35) \approx 0.0436$

5. Probability $P(X < 30)$: Using a binomial distribution calculator: $P(X < 30) \approx 0.5616$

Summary of Results

1. Mean $\mu = 28.91$
2. Variance $\sigma^2 \approx 14.7861$ (rounded to 4 decimal places)
3. Standard Deviation $\sigma = 3.8431$ (rounded to 4 decimal places)
4. Probability $P(X > 35) \approx 0.0436$
5. Probability $P(X < 30) \approx 0.5616$

Would you like details on any specific part of the solution? Here are some follow-up questions:

1. How do you interpret the mean and variance in the context of a binomial distribution?
2. What is the significance of the standard deviation in this scenario?
3. Can you explain how to use the binomial probability formula?
4. What assumptions do we make when modeling a situation with a binomial distribution?
5. How would the results change if $p$ were to increase or decrease significantly?

Tip: When dealing with binomial distributions, it can be helpful to visualize the probability distribution to better understand the likelihood of different outcomes.