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In probability and statistics, a gamma distribution with a large shape parameter \( \alpha \) can often be approximated by a normal distribution. This is an incredibly useful trick because the normal distribution allows for simpler calculations and an easier interpretation of data. The gamma distribution itself is defined by two parameters: the shape parameter \( \alpha \) and the rate parameter \( \beta \). Here, \( \alpha = 50 \) and \( \beta = 2 \).

When \( \alpha \) is large, the distribution's shape tends to resemble that of a normal distribution. This is due to the central limit theorem, which suggests that if you add up a large number of independent random variables with the same distribution, the sum will tend to follow a normal distribution. The normal distribution approximation allows us to utilize the mean \( \mu = \alpha / \beta \) and the variance \( \sigma^2 = \alpha / \beta^2 \) derived from the gamma distribution, but then to proceed with the normal distribution's easier probability calculations.
For the specific example, given \( \alpha = 50 \) and \( \beta = 2 \), the mean is \( 25 \) and the variance is \( 12.5 \). This substantially simplifies the problem since normal distributions are notorious for having calculable probabilities through Z-scores, as opposed to the more complex gamma calculations.

The concept of a standard normal variable plays a pivotal role in probability calculations. In any normal distribution, a standard normal variable, denoted as \( Z \), is used to help find probabilities associated with specific outcomes. This is done by converting any normal random variable into a standard normal variable through the formula: \[ Z = \frac{X - \mu}{\sigma} \]Here, \( X \) is a random variable, \( \mu \) is the mean of the distribution, and \( \sigma \) is the standard deviation.

In our case, the task is to know the likelihood of a student spending at most 125 hours on the project. Thus, \( X = 125 \), \( \mu = 25 \), and \( \sigma \approx 2.5 \). Plugging these into the formula helps convert \( X \) into a standard normal variable, giving: \[ Z = \frac{125 - 25}{2.5} = 40 \]This transformation allows us to then use standard resources like Z-tables or calculators to determine the probability associated with \( Z = 40 \). A \( Z \) value of 40 is extremely high, which implies almost certainty in terms of probability calculations.

Once a problem has been reduced to calculating the probability of a standard normal variable, using a normal distribution table or a calculator is straightforward.

A Z-table lists the cumulative probabilities of a standard normal distribution. It is a vital tool for anyone working with normal distributions, as it allows one to find the probability that a random variable falls below a certain threshold. In our specific scenario, after calculating a \( Z \) value of 40, the next step is to recognize that a Z-score that large falls well beyond the range typically covered in such tables. Most tables only go up to 3 or 4.
For practical purposes, a \( Z \) value of 40 means that the probability, \( P(X \leq 125) \), is effectively 1. This indicates almost complete certainty that a randomly selected student would spend less than or equal to 125 hours on the project. This simplification demonstrates the power of converting complex distributions into standard forms, as it streamlines computation and highlights how extreme values influence probability.