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 between 0.5 kg and 2.5 kg during freshman year.

Verified step by step guidance

1. Step 1: Identify the given parameters. The problem states that the weight gain is normally distributed with a mean (μ) of 1.2 kg and a standard deviation (σ) of 4.9 kg. We are tasked with finding the probability that a randomly selected male college student gains between 0.5 kg and 2.5 kg.
2. Step 2: Standardize the values using the z-score formula. The z-score formula is given by: $z=\frac{x-\mu }{\sigma }$, where x is the value of interest, μ is the mean, and σ is the standard deviation. Compute the z-scores for x = 0.5 and x = 2.5.
3. Step 3: Use the z-scores to find the cumulative probabilities. Once the z-scores are calculated, use a standard normal distribution table or a statistical software to find the cumulative probabilities corresponding to these z-scores.
4. Step 4: Subtract the cumulative probability of the lower z-score (corresponding to x = 0.5) from the cumulative probability of the higher z-score (corresponding to x = 2.5). This will give the probability that the weight gain is between 0.5 kg and 2.5 kg.
5. Step 5: Interpret the result. The final probability represents the likelihood that a randomly selected male college student gains between 0.5 kg and 2.5 kg during their freshman year. Ensure the result is reasonable given the context of the problem.

Key Concepts

Normal Distribution

Normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. In this context, the weight gain of male college students follows a normal distribution with a specified mean and standard deviation, allowing us to use statistical methods to calculate probabilities related to weight gain.

Central Limit Theorem (CLT)

The Central Limit Theorem states that the sampling 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 when dealing with sample means and allows for the application of normal distribution properties even when the sample size is small, provided the population is normally distributed.

Probability Calculation

Probability calculation involves determining the likelihood of a specific event occurring within a defined range. In this scenario, we need to calculate the probability that a randomly selected male college student gains between 0.5 kg and 2.5 kg, which requires using the properties of the normal distribution to find the area under the curve between these two values.