Using the Central Limit Theorem. In Exercises 5–8, assume that the amounts of weight that male college students gain during their freshman year are normally distributed with a mean of 1.2 kg and a standard deviation of 4.9 kg (based on Data Set 13 “Freshman 15” in Appendix B).


a. If 1 male college student is randomly selected, find the probability that he gains at least 2.0 kg during his freshman year..)

Verified step by step guidance

1. Step 1: Identify the given parameters. From the problem, the mean (μ) is 1.2 kg, the standard deviation (σ) is 4.9 kg, and we are looking for the probability that a randomly selected male college student gains at least 2.0 kg. This means we need to calculate P(X ≥ 2.0), where X is the weight gain.
2. Step 2: Standardize the value of 2.0 kg using the z-score formula. The z-score formula is given by: $z\; =\; (X\; -\; \mu )\; /\; \sigma$. Substitute the values X = 2.0, μ = 1.2, and σ = 4.9 into the formula.
3. Step 3: Simplify the z-score calculation to find the standardized z-value. This will give you the z-score corresponding to the weight gain of 2.0 kg.
4. Step 4: Use the standard normal distribution table (or a statistical software/tool) to find the cumulative probability corresponding to the calculated z-score. This cumulative probability represents P(X ≤ 2.0).
5. Step 5: Since the problem asks for P(X ≥ 2.0), use the complement rule: P(X ≥ 2.0) = 1 - P(X ≤ 2.0). Subtract the cumulative probability from 1 to find the desired probability.

Key Concepts

Central Limit Theorem

The Central Limit Theorem (CLT) states that the distribution of the sample mean will approach a normal distribution as the sample size increases, regardless of the original distribution of the population. This theorem is crucial for making inferences about population parameters based on sample statistics, especially when dealing with large samples.

Normal Distribution

A normal distribution is a continuous probability distribution characterized by its bell-shaped curve, defined by its mean and standard deviation. In this context, the weights gained by male college students are normally distributed, which allows us to use properties of the normal distribution to calculate probabilities related to weight gain.

Z-Score

A Z-score is a statistical measurement that describes a value's relationship to the mean of a group of values, expressed in terms of standard deviations. It is calculated by subtracting the mean from the value and dividing by the standard deviation. Z-scores are essential for determining probabilities in a normal distribution, allowing us to find the likelihood of a specific weight gain.