Sampling distributions are crucial for understanding how sample statistics behave. For means, the distribution centers on the population mean, with standard error decreasing as sample size grows. The Central Limit Theorem ensures normality for large samples.
For proportions, the sampling distribution also centers on the population value. The standard error depends on both the proportion and sample size. Normal approximation applies when samples are large enough, enabling z-score calculations for probabilities and inference.
Sampling Distribution of the Sample Mean
Mean and standard deviation of sampling distributions
- Mean of sampling distribution equals population mean (μ) serves as unbiased estimator
- Standard error of mean calculated as $SE = \frac{\sigma}{\sqrt{n}}$ decreases with larger sample sizes (n)
- Central Limit Theorem applies to sample means from any population distribution approximates normal for large n
- Larger samples narrow distributions leading to smaller standard errors
Probabilities in sampling distributions
- Z-score calculation $Z = \frac{\bar{X} - \mu}{\sigma / \sqrt{n}}$ standardizes sample mean
- Standard normal distribution has mean 0, standard deviation 1, symmetric bell shape
- Probability calculations use z-table or calculator for area under curve interpreted in context
- Probability questions involve sample mean above/below value, between two values, finding critical values for confidence intervals
Sampling Distribution of the Sample Proportion
Sampling distribution of sample proportions
- Mean equals population proportion (p) serves as unbiased estimator
- Standard error calculated as $SE_p = \sqrt{\frac{p(1-p)}{n}}$ depends on population proportion and sample size
- Normality conditions require large sample size (n ≥ 30), np ≥ 5 and n(1-p) ≥ 5
- Larger samples decrease standard error, distribution becomes more normal-shaped
Normal approximation for large samples
- Binomial distribution models successes in fixed trials (coin flips, product defects)
- Normal approximation conditions: n ≥ 30, np ≥ 5 and n(1-p) ≥ 5
- Continuity correction adjusts discrete to continuous by adding/subtracting 0.5
- Z-score for proportions: $Z = \frac{\hat{p} - p}{\sqrt{\frac{p(1-p)}{n}}}$
- Applied in hypothesis testing, confidence intervals for proportions
- Not suitable for small samples or extreme proportions (very high or low)