6.3 Sampling Distribution of the Mean

When we take samples from a population, their means form a distribution. This sampling distribution of the mean tells us how sample averages behave. It's key for estimating population parameters and making inferences about large groups from smaller samples.

The standard error measures the spread of this distribution. It shrinks as sample size grows, making larger samples more precise. We use this to build confidence intervals, helping us gauge how close our sample mean is to the true population average.

Sampling Distribution of the Mean

Sampling distribution of the mean

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• Probability distribution of sample means obtained from repeated sampling of a population
• Represents distribution of all possible sample means for a given sample size (e.g., means of samples with n=30)
• Key characteristics:
  • Shape: Approaches normal distribution as sample size increases (Central Limit Theorem)
  • Center: Mean equals population mean ($\mu$)
  • Spread: Standard deviation (standard error) equals population standard deviation ($\sigma$) divided by square root of sample size ($\sqrt{n}$)

Standard error computation and interpretation

• Standard error of the mean ($\sigma_{\bar{x}}$): Standard deviation of sampling distribution of the mean
  • Formula: $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$, where $\sigma$ is population standard deviation and $n$ is sample size
• Relationship with sample size:
  • Larger sample sizes decrease standard error (e.g., n=100 has smaller standard error than n=30)
  • Larger samples result in smaller spread of sampling distribution, making sample means more precise estimates of population mean
• Relationship with population standard deviation:
  • Standard error directly proportional to population standard deviation
  • Larger population standard deviation results in larger standard error, indicating more variability in sample means (e.g., $\sigma=10$ yields larger standard error than $\sigma=5$)

Confidence intervals for population mean

• Range of values likely to contain population mean with certain confidence level
• Constructing confidence interval:
  1. Determine desired confidence level (e.g., 95%) and find corresponding z-score from standard normal distribution
  2. Calculate standard error using formula $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$
  3. Calculate margin of error by multiplying z-score by standard error
  4. Add and subtract margin of error from sample mean to obtain lower and upper bounds of confidence interval
• Interpretation:
  • 95% confidence interval means if repeated samples were taken and intervals constructed, approximately 95% would contain true population mean (e.g., 95 out of 100 intervals)

Sample size for mean estimation

• Determining minimum sample size to estimate population mean:
  1. Specify desired confidence level (e.g., 99%) and corresponding z-score
  2. Determine acceptable margin of error (e.g., $\pm 3$ units)
  3. Estimate or know population standard deviation ($\sigma$)
  4. Use formula: $n = (\frac{z \cdot \sigma}{E})^2$, where $n$ is minimum sample size, $z$ is z-score, $\sigma$ is population standard deviation, and $E$ is margin of error
• Larger sample size required when:
  • Higher confidence level desired (e.g., 99% requires larger n than 90%)
  • Smaller margin of error needed (e.g., $\pm 1$ unit requires larger n than $\pm 5$ units)
  • Population standard deviation larger (e.g., $\sigma=20$ requires larger n than $\sigma=5$)