6. Normal Distribution and Continuous Random Variables
Standard Normal Distribution
2:31 minutesProblem 6.4.6a
Textbook Question
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 guidance1
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.
Step 2: Standardize the value of 2.0 kg using the z-score formula. The z-score formula is given by: . Substitute the values X = 2.0, μ = 1.2, and σ = 4.9 into the formula.
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.
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).
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.
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
2mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
Recommended video:
Guided course
04:52
Calculating the Mean
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.
Recommended video:
Guided course
09:47Finding Standard Normal Probabilities using z-Table
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.
Recommended video:
Guided course
06:31Z-Scores From Given Probability - TI-84 (CE) Calculator
9:47mWatch next
Master Finding Standard Normal Probabilities using z-Table with a bite sized video explanation from Patrick
Start learning
