6. Normal Distribution and Continuous Random Variables
Standard Normal Distribution
2:32 minutesProblem 6.4.5a
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 has no weight gain during his freshman year. (That is, find the probability that during his freshman year, his weight gain is less than or equal to 0 kg.)
Verified step by step guidance1
Step 1: Identify the given parameters for the normal distribution. The mean (μ) is 1.2 kg, and the standard deviation (σ) is 4.9 kg. The random variable X represents the weight gain of a male college student during his freshman year.
Step 2: Define the probability to be calculated. We are tasked with finding the probability that the weight gain is less than or equal to 0 kg, i.e., P(X ≤ 0).
Step 3: Standardize the random variable X to convert it into a standard normal variable Z using the formula Z = (X - μ) / σ. Substituting the values, Z = (0 - 1.2) / 4.9.
Step 4: Simplify the Z-score calculation to find the standardized value. This will give you the Z-score corresponding to X = 0.
Step 5: Use a standard normal distribution table or a statistical software to find the cumulative probability corresponding to the calculated Z-score. This cumulative probability represents P(X ≤ 0).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Normal Distribution
Normal distribution is a probability distribution that is symmetric about the mean, depicting 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.
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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 population distribution is not normal, provided the sample size is sufficiently large.
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Probability Calculation
Probability calculation involves determining the likelihood of a specific event occurring within a defined set of outcomes. In this scenario, we need to calculate the probability that a randomly selected male college student has a weight gain of less than or equal to 0 kg, which requires using the properties of the normal distribution to find the corresponding z-score and then referencing the standard normal distribution table.
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