Math Problem Statement
Sampling distribution from a normal population
Solution
A sampling distribution refers to the probability distribution of a statistic (e.g., mean, variance, proportion) based on a random sample from a population. When sampling from a normal population, the sampling distribution has specific properties depending on the statistic being considered and the sample size. Here's a breakdown:
1. Sampling Distribution of the Sample Mean
- If the population is normally distributed with mean and standard deviation , then:
- The sample mean is also normally distributed.
- The mean of the sampling distribution is .
- The standard deviation of the sampling distribution (known as the standard error) is: where is the sample size.
- Formula:
2. Central Limit Theorem (CLT)
- Even if the population is not normal, for a sufficiently large sample size (), the sampling distribution of the sample mean approaches a normal distribution.
- This allows us to apply normal distribution properties even to non-normally distributed populations when is large.
3. Sampling Distribution of the Sample Variance
- When sampling from a normal population, the sample variance has a distribution related to the chi-square distribution: where represents the degrees of freedom.
4. Sampling Distribution of the Sample Proportion
- If sampling involves proportions (e.g., number of successes in a sample of size ), and the population is normal or approximated as normal, the sampling distribution of the sample proportion is: where is the population proportion.
Example Use:
If a population has and , and you take a sample of size , then:
- The sample mean has and .
Would you like more detailed examples, visualizations, or an application of these principles?
Related Questions:
- How does the Central Limit Theorem affect the sampling distribution for non-normal populations?
- What is the role of standard error in hypothesis testing?
- How do sample size and variability affect the shape of the sampling distribution?
- What happens to the sampling distribution as the sample size increases?
- How do we calculate probabilities using the sampling distribution of the sample mean?
Tip:
Always verify whether the sample size is large enough for approximations (like CLT) when dealing with non-normal populations.
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Math Problem Analysis
Mathematical Concepts
Probability Distribution
Sampling Distribution
Central Limit Theorem
Chi-Square Distribution
Formulas
Standard error: σ_x̄ = σ / √n
Sample variance: (n-1)s^2 / σ^2 ~ χ²_(n-1)
Sample proportion: p̂ ~ N(p, √(p(1-p)/n))
Theorems
Central Limit Theorem
Suitable Grade Level
Undergraduate Level (Statistics or Probability)
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