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Questions and Answers
What is the mean of the sampling distribution of the means if the sample size is sufficiently large?
What is the mean of the sampling distribution of the means if the sample size is sufficiently large?
What is the effect of increasing the sample size on the standard deviation of the sampling distribution of the means?
What is the effect of increasing the sample size on the standard deviation of the sampling distribution of the means?
What is the variance of the sampling distribution of the means if the sample size is 4 and the variance of the population is 4?
What is the variance of the sampling distribution of the means if the sample size is 4 and the variance of the population is 4?
What is the shape of the sampling distribution of the means if the sample size is large enough?
What is the shape of the sampling distribution of the means if the sample size is large enough?
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What is the standard deviation of the sampling distribution of the means if the sample size is 6 and the standard deviation of the population is 3.2?
What is the standard deviation of the sampling distribution of the means if the sample size is 6 and the standard deviation of the population is 3.2?
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What is the purpose of the central limit theorem?
What is the purpose of the central limit theorem?
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What is the implication of the term 'with replacement' in Central Limit Theorem?
What is the implication of the term 'with replacement' in Central Limit Theorem?
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What is the purpose of constructing the sampling distribution in the Central Limit Theorem?
What is the purpose of constructing the sampling distribution in the Central Limit Theorem?
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What is the shape of the sampling distribution of x̄ when n = 2?
What is the shape of the sampling distribution of x̄ when n = 2?
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What is the mean of the sampling distribution of the means when the population mean is μ = 7 and the sample size is n = 9?
What is the mean of the sampling distribution of the means when the population mean is μ = 7 and the sample size is n = 9?
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What is the implication of the term 'without replacement' in Central Limit Theorem?
What is the implication of the term 'without replacement' in Central Limit Theorem?
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What is the purpose of computing the mean, variance, and standard deviation of the sampling distribution of the means?
What is the purpose of computing the mean, variance, and standard deviation of the sampling distribution of the means?
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Study Notes
Central Limit Theorem (CLT)
- The CLT states that the frequency distribution of the sample mean (X̄) has the following properties:
- Mean of the sample mean is equal to the mean of the population (𝝁)
- Variance of the sampling distribution of means is equal to the variance of the population divided by the sample size (n)
- Standard deviation of the sampling distribution of means is equal to the standard deviation of the population divided by the square root of the sample size (√n)
- The sampling distribution of the mean will always be normally distributed, as long as the sample size is large enough
Examples and Exercises
- Example 1:
- Given a population with mean (𝜇) = 4.6 and variance (𝝈²) = 4, and samples of size n = 4,
- The mean of the sampling distribution of the means is equal to the population mean (𝝁), and the standard deviation is equal to the population standard deviation divided by √n
- Example 2:
- Given a population with mean (𝜇) = 3 and standard deviation (𝝈) = 1.8, and samples of size n = 2,
- The mean of the sampling distribution of the means is equal to the population mean (𝝁), and the variance of the sampling distribution of the means is equal to the population variance divided by n
- Exercise 1:
- Given a population with mean (𝜇) = 5 and standard deviation (𝝈) = 3.2, and samples of size n = 6,
- Find the mean and variance of the population where the sampling distribution is derived
- Activity 1:
- Given a population with mean (𝜇) = 7 and variance (𝝈²) = 4, and samples of size n = 9,
- Find the mean and standard deviation of the sampling distribution of the means
Central Limit Theorem (with Replacement)
- The term "with replacement" implies that the samples can be selected repeatedly in random sampling
- To find the mean, variance, and standard deviation of the sampling distribution of the means:
- Compute the mean, variance, and standard deviation of the sampling distribution of the means
- Construct the sampling distribution
- Draw the probability histogram of the sampling distribution of the means
Example (with Replacement)
- Consider all possible samples of size n = 2 that can be drawn with replacement from the population 1, 2, 3, and 4
- The sampling distribution of the mean is approximately normally distributed
Central Limit Theorem (without Replacement)
- The term "without replacement" implies that no samples can be selected repeatedly in random sampling
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Description
Test your understanding of the Central Limit Theorem and its properties, including the mean, variance, and standard deviation of the sampling distribution of means. Learn how the sample size affects the variance and standard deviation of the sampling distribution.
