Assume that the amounts of weight that male college students gain during their freshman year are normally distributed with a mean of mu equals 1.1 kg and a standard deviation of sigma equals 4.6 kg. Complete parts​ (a) through​ (c) below.

a. If 1 male college student is randomly​ selected, find the probability that he gains between 0 kg and 3 kg during freshman year.

The probability is

To solve this, we will use the properties of the normal distribution.

Given:
- Mean \( \mu = 1.1 \, \text{kg} \)
- Standard deviation \( \sigma = 4.6 \, \text{kg} \)

We are asked to find the probability that a randomly selected male college student gains between 0 kg and 3 kg during their freshman year. This is equivalent to finding the probability that \( X \) (the weight gain) is between 0 and 3, i.e., \( P(0 \leq X \leq 3) \).

We can standardize this using the **z-score** formula:

\[
z = \frac{x - \mu}{\sigma}
\]

### Step 1: Find the z-scores for 0 kg and 3 kg.

- For \( X = 0 \, \text{kg} \):
  \[
  z_0 = \frac{0 - 1.1}{4.6} = \frac{-1.1}{4.6} \approx -0.2391
  \]
  
- For \( X = 3 \, \text{kg} \):
  \[
  z_3 = \frac{3 - 1.1}{4.6} = \frac{1.9}{4.6} \approx 0.4130
  \]

### Step 2: Use the standard normal distribution table or a calculator to find the probabilities corresponding to these z-scores.

- \( P(z_0) \approx 0.4052 \) (probability corresponding to \( z_0 = -0.2391 \))
- \( P(z_3) \approx 0.6604 \) (probability corresponding to \( z_3 = 0.4130 \))

### Step 3: Calculate the probability.

The probability that the student gains between 0 kg and 3 kg is:

\[
P(0 \leq X \leq 3) = P(z_3) - P(z_0) = 0.6604 - 0.4052 = 0.2552
\]

So, the probability is approximately **0.2552**.

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Would you like further clarification, or any details on this process? 

Here are 5 additional questions based on this problem:
1. How would the probability change if the standard deviation was larger?
2. What is the probability that a student gains more than 3 kg during their freshman year?
3. What z-score corresponds to a weight gain of 5 kg?
4. If we randomly select 5 students, how would the probability of their average weight gain being between 0 and 3 kg change?
5. How does the assumption of normal distribution affect the interpretation of this result?

**Tip:** The z-score standardizes values, allowing us to compare different normal distributions using the standard normal (mean 0, standard deviation 1).

Calculate the probability that a male college student gains between 0 kg and 3 kg during freshman year using the properties of normal distribution. The problem involves a mean weight gain of 1.1 kg and a standard deviation of 4.6 kg. Learn step-by-step how to use z-scores to solve this problem.

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