To solve this problem, we need to calculate the probability that a male college student gains between 0 kg and 3 kg during freshman year. Here’s the plan:

1. Understand the Normal Distribution:

   • Mean ($\mu$) = 1.3 kg
   • Standard Deviation ($\sigma$) = 4.7 kg

2. Standardize the Values: Using the Z-score formula: $Z = \frac{X - \mu}{\sigma}$

   • For $X = 0$ kg: $Z_0 = \frac{0 - 1.3}{4.7} = -0.2766$
   • For $X = 3$ kg: $Z_3 = \frac{3 - 1.3}{4.7} = 0.3617$

3. Find the Probabilities Corresponding to the Z-Scores: Using the Z-table or a statistical calculator:

   • For $Z_0 = -0.2766$: $P(Z < -0.2766) \approx 0.3910$
   • For $Z_3 = 0.3617$: $P(Z < 0.3617) \approx 0.6410$

4. Calculate the Desired Probability: The probability of gaining between 0 kg and 3 kg is: $P(0 \leq X \leq 3) = P(Z < 0.3617) - P(Z < -0.2766)$ $P(0 \leq X \leq 3) = 0.6410 - 0.3910 = 0.2500$

Final Answer:

The probability is 0.2500.

Would you like a step-by-step explanation of any part of this process? Here are some related questions to expand on this topic:

1. How do you find probabilities for values outside a certain range in a normal distribution?
2. What is the purpose of standardizing values with Z-scores?
3. How does the shape of the normal distribution affect probabilities?
4. What is the cumulative distribution function (CDF) for a normal distribution?
5. Can you calculate probabilities using a calculator instead of a Z-table?

Tip: Always check if the data follows a normal distribution before applying these methods.