Unit 5: Sampling Distributions Notes

Normal Distribution Calculations

• Normal distribution: A model for how sample statistics vary under repeated sampling.
• Z-scores: Measure standard deviations from the mean. Used in calculations involving normal distributions.
• TI-84 Calculations:
  • normalcdf(lowerbound, upperbound): Gives area (probability) between two z-scores.
  • invNorm(area): Gives z-score with given area to the left.
  • Can work with raw scores using mean and standard deviation: normalcdf(lowerbound, upperbound, mean, standard deviation).

Example 5.1

• Problem: Calculate probabilities based on light bulb life expectancy (mean = 1500 hours, SD = 75 hours).
• Solution: Use z-scores and normalcdf to find probabilities.
• Full credit requires noting distribution, parameters, boundaries, values of interest, and correct probability.

Central Limit Theorem (CLT)

• Central Limit Theorem: For large sample sizes (n ≥ 30), sample means are approximately normally distributed.
  • Mean of sample means = population mean.
  • Standard deviation of sample means = population SD / sqrt(n).
• Key ideas:
  • Averages vary less than individual values.
  • Larger samples lead to less variance.
  • If the population is normal, so is the sampling distribution.

Example 5.2

• Problem: Probability calculations for life expectancy of naked mole rats (mean = 21 years, SD = 3 years, n = 40).
• Solution: Use CLT and normalcdf for probability calculations.

Biased and Unbiased Estimators

• Bias: Occurs when sampling distribution is not centered on population parameter.
• Unbiased Estimators: Sample proportions, means, and slopes centered on population values.
• Example 5.3: Evaluating estimators for consistency with population parameters.
  • Solutions involve checking for unbiasedness and variability.

Sampling Distribution for Sample Proportions

• Proportion Calculation: Interested in presence/absence of attribute.
• Normal Approximation: For large n, sampling distribution is approximately normal.

Example 5.4

• Problem: Probability of sample proportion with math anxiety experiencing pain-like brain activity.
• Solution: Use normal approximation and normalcdf.

Sampling Distribution for Differences in Sample Proportions

• Differences: Mean of differences is difference of means; variance is sum of variances.

Example 5.5

• Problem: Probability of difference in eating habits in revamped vs unrevamped restaurants.
• Solution: Use independent samples and normal approximation for differences.

Sampling Distribution for Sample Means

• Variance: Variance of sample means = variance of sums divided by n².

Example 5.6

• Problem: Expected caffeine content from energy drink samples.
• Solution: Calculate mean and standard deviation of sample means.

Sampling Distribution for Differences in Sample Means

• Differences: Variance of differences is sum of individual variances.

Example 5.7

• Problem: Genetic mutations from different age groups of new fathers.
• Solution: Use sampling distribution for differences for probability calculations.

Simulation of a Sampling Distribution

• Simulation: Used for non-normal statistics to understand sampling distributions.
• Example: Simulated distribution of the number of dreams remembered by students.
  • Results showed roughly bell-shaped distribution of medians, skewed variances, and other distributions.