Unit 5: Sampling Distributions

Normal Distribution Calculations

• Normal distribution is a model for how many sample statistics vary from a population.
• Calculations often use z-scores, representing standard deviations from the mean.
• TI-84 calculator:
  • normalcdf(lowerbound, upperbound) gives probability between z-scores.
  • invNorm(area) gives z-score for a given probability.
• Example 5.1: Calculating probabilities for lightbulb life expectancy using z-scores.

Central Limit Theorem (CLT)

• CLT: For a large sample size (n ≥ 30), the distribution of sample means is approximately normal.
  • Sample mean equals population mean.
  • Sample mean's standard deviation is population standard deviation divided by sqrt(n).
• Key ideas:
  • Averages vary less than individual values.
  • Larger samples result in less variation.
• Example 5.2: Probability of life expectancy in samples of naked mole rats.

Biased and Unbiased Estimators

• Bias: Sampling distribution not centered on the population parameter.
• Unbiased estimators (proportions, means, slopes) are centered on population values.
• Example 5.3: Evaluating estimators in manufacturing quality control.

Sampling Distribution for Sample Proportions

• Relates to qualitative attributes (presence/absence).
• For large n, distribution appears normal.
• Example 5.4: Probability calculation for math anxiety and brain activity.

Sampling Distribution for Differences in Sample Proportions

• Compares differences in sample proportions from two populations.
• Mean of differences equals the difference of means.
• Example 5.5: Examining differences in eating habits between two restaurant settings.

Sampling Distribution for Sample Means

• The variance of sample means is the population variance divided by n.
• Example 5.6: Mean and standard deviation of caffeine in energy drinks over samples.

Sampling Distribution for Differences in Sample Means

• Variance of differences equals the sum of individual variances.
• Example 5.7: Genetic mutations in children of different age fathers.

Simulation of a Sampling Distribution

• Used when normal distribution isn't sufficient.
• Example simulation of dream recall in students shows different distribution shapes for medians, variances, and minimums.

• To receive full credit for probability calculations, show:
  1. Name of the distribution
  2. Parameters
  3. Boundary values
  4. Values of interest
  5. Correct probability