The Central Limit Theorem for sums is a powerful statistical tool that helps us understand the behavior of large samples. It states that the sum of many independent random variables tends to follow a normal distribution, regardless of the original population's shape.

This theorem is crucial for making predictions and inferences about populations. It allows us to calculate probabilities, determine percentiles, and test hypotheses using z-scores, even when dealing with non-normal distributions in large samples.

The Central Limit Theorem for Sums

Central Limit Theorem for sums

• States sampling distribution of sum of large number of independent and identically distributed (i.i.d.) random variables will be approximately normal, regardless of shape of population distribution
• Applies to sum of random variables, not just mean
• As sample size (n) increases, distribution of sum of random variables becomes closer to normal distribution (convergence in distribution)
  • Larger sample size, better normal approximation
• Mean of sampling distribution of sum equals sum of population means: $\mu_{\sum X} = n\mu$
  • $n$ = sample size
  • $\mu$ = population mean
• Standard deviation of sampling distribution of sum equals population standard deviation multiplied by square root of sample size: $\sigma_{\sum X} = \sqrt{n}\sigma$
  • $\sigma$ = population standard deviation

Mean and standard deviation of sampling distributions

• Calculate mean of sampling distribution for sums using formula: $\mu_{\sum X} = n\mu$
  • $n$ = sample size
  • $\mu$ = population mean
• Calculate standard deviation of sampling distribution for sums using formula: $\sigma_{\sum X} = \sqrt{n}\sigma$
  • $n$ = sample size
  • $\sigma$ = population standard deviation
• Example: Population mean = 10, population standard deviation = 2, sample size = 25
  • $\mu_{\sum X} = 25 \times 10 = 250$
  • $\sigma_{\sum X} = \sqrt{25} \times 2 = 5 \times 2 = 10$

Z-scores in Central Limit Theorem

• Z-score formula for sums: $z = \frac{\sum X - \mu_{\sum X}}{\sigma_{\sum X}}$
  • $\sum X$ = sum of random variables in sample
  • $\mu_{\sum X}$ = mean of sampling distribution for sums
  • $\sigma_{\sum X}$ = standard deviation of sampling distribution for sums
• Z-score represents number of standard deviations a sum is away from mean of sampling distribution for sums
• When sample size is large enough, sampling distribution for sums is approximately normal
• Z-score can be used to:
  1. Calculate probabilities using standard normal distribution table
  2. Determine percentiles and quantiles (90th percentile, lower quartile)
  3. Test hypotheses about population sum (null hypothesis, alternative hypothesis)
• Standardization of sums allows for comparison across different probability distributions

Statistical Foundations and Applications

• Law of large numbers: As sample size increases, sample mean converges to population mean
• Probability distribution: Describes likelihood of different outcomes in an experiment
• Statistical inference: Process of drawing conclusions about populations based on sample data
• Central Limit Theorem for sums is crucial for making inferences about population parameters