91Ó°ÊÓ

The gamma distribution is a continuous probability distribution often used in scenarios where the process involves waiting times or time until an event happens, among others. It's characterized by two parameters—shape (\(\alpha\)) and rate (\(\beta\)). The shape parameter (\(\alpha\)) affects the distribution's form, while the rate parameter (\(\beta\)) scales the distribution along the x-axis.In practical terms:

• The mean of a gamma distribution is calculated as \(\mu = \alpha \cdot \beta\).
• The variance is given by \(\sigma^2 = \alpha \cdot \beta^2\).

In the context of the exercise, these formulas are used to derive the parameters needed for a normal approximation when the shape parameter is large. This makes the gamma distribution versatile and able to closely model normal distributions under certain conditions.

The Central Limit Theorem (CLT) is a fundamental concept in statistics. It states that, under certain conditions, the sum or average of a large number of independent random variables will be approximately normally distributed, regardless of the original distribution of the variables.### Application of CLTIn the exercise at hand, the large value of the shape parameter \(\alpha = 50\) allows us to invoke the Central Limit Theorem. This lets us approximate the gamma distribution using a normal distribution with the same mean and variance. This is particularly useful because many statistical methods are easier to apply to normal distributions.This approximation helps streamline complex computations, enabling us to use normal distribution tools, like z-score tables, for calculations. Without CLT, approximating distributions like gamma in simpler forms would be much tougher, especially when dealing with large datasets or parameters.

Z-score calculation is a critical step in many statistical analyses, including this exercise. A z-score represents how many standard deviations an element \(X\) is from the mean \(\mu\). The formula for z-score is:\[ z = \frac{X - \mu}{\sigma} \]Where:

• \(X\) is the value in question, in this case, 125 hours.
• \(\mu\) is the mean, previously calculated as 100.
• \(\sigma\) is the standard deviation, calculated as \(\sqrt{200}\)

In our exercise, plugging the numbers into the formula provides us with a z-score of approximately 1.77. This score references where the specific time falls in relation to the entire distribution, thereby allowing us to use the standard normal distribution tables to find the corresponding probability of students spending up to that many hours on the project.

Probability theory is the branch of mathematics concerned with analyzing random phenomena. It provides a framework for reasoning about uncertain events happening in real life or any scientific data. Within the exercise, probability theory enables the transformation of situations described by the gamma distribution into a normal approximation to make problem-solving more feasible.### Probability Calculations in PracticeIn the scenario given, once the gamma distribution is approximated to a normal one via the Central Limit Theorem, probability theory allows us to estimate the likelihood of a student spending less than or up to 125 hours on the project. By using the derived z-score, we can use statistical tables or software to find:

• \(P(Z \leq 1.77) \approx 0.9616\), which signifies a 96.16% chance that a student spends at most 125 hours on the project.

This way, probability theory not only simplifies complex distributions but also inputs logical numbers into uncertain scenarios, providing clarity and actionable insights.