6. Normal Distribution and Continuous Random Variables
Standard Normal Distribution
3:49 minutesProblem 6.4.7b
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).
b. If 9 male college students are randomly selected, find the probability that their mean weight gain during freshman year is between 0 kg and 3 kg.
Verified step by step guidance1
Step 1: Identify the given parameters. The population mean (μ) is 1.2 kg, the population standard deviation (σ) is 4.9 kg, and the sample size (n) is 9. We are tasked with finding the probability that the sample mean (̄) is between 0 kg and 3 kg.
Step 2: Use the Central Limit Theorem to determine the sampling distribution of the sample mean. According to the theorem, the sampling distribution of the sample mean is normally distributed with mean μ and standard error (SE) given by SE = σ / √n. Calculate SE using the formula: .
Step 3: Standardize the sample mean values (0 kg and 3 kg) to z-scores using the formula: . Compute the z-scores for both 0 kg and 3 kg.
Step 4: Use the standard normal distribution table (or a statistical software) to find the cumulative probabilities corresponding to the z-scores calculated in Step 3. These cumulative probabilities represent the area under the standard normal curve up to the respective z-scores.
Step 5: Subtract the cumulative probability for the lower z-score (corresponding to 0 kg) from the cumulative probability for the higher z-score (corresponding to 3 kg). This difference gives the probability that the sample mean weight gain is between 0 kg and 3 kg.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Central Limit Theorem (CLT)
The Central Limit Theorem 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 is crucial for making inferences about population parameters based on sample statistics, especially when dealing with means.
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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 weight gain of male college students is normally distributed, which allows us to use properties of the normal distribution to calculate probabilities related to sample means.
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Sampling Distribution of the Mean
The sampling distribution of the mean is the probability distribution of all possible sample means from a population. For a sample size of 9, the mean of this distribution will equal the population mean, while its standard deviation (standard error) is the population standard deviation divided by the square root of the sample size, allowing us to find probabilities for the sample mean.
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