7.3: Using the Central Limit Theorem

Theoretical Foundations: Law of Large Numbers and Central Limit Theorem

• The Law of Large Numbers:

  • States that as the sample size increases, the sample mean $\bar{x}$ tends to approach the population mean $\mu$.

• The Central Limit Theorem (CLT):

  • States that as sample size $n$ gets larger, the distribution of sample means tends to follow a normal distribution pattern.

  • As $n$ gets larger, the standard deviation of that normal distribution becomes progressively smaller according to the formula: $\sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}}$.

• Theoretical Connection:

  • Because the standard deviation of the sampling distribution decreases toward zero as $n$ approaches infinity, the sample mean converges directly to the population mean $\mu$. Consequently, the Central Limit Theorem proves the Law of Large Numbers.

Selection Framework: Rules-of-Thumb for Problem Types

• Single Value Inquiries (Non-CLT Application):

  • When asked to determine information about a single individual value drawn from a distribution, do not apply either version of the Central Limit Theorem.

  • Utilize the specific properties and parameters of the original distribution directly.

  • Care must be taken to apply the appropriate distribution information corresponding to that specific single value.

• Sample Average Inquiries (CLT for Averages):

  • When asked to calculate probabilities or values related to the average (mean) of a sample, use the Central Limit Theorem for averages.

  • Requires a sufficiently large sample size $n$.

• Sample Sum Inquiries (CLT for Sums):

  • When asked to calculate probabilities or values related to the sum of a sample, use the Central Limit Theorem for sums.

  • Requires a sufficiently large sample size $n$.

Illustrative Applications and Problem Scenarios

• Sample Average Calculation Problem:

  • Scenario: A sample of $36$ numbers is drawn from an underlying distribution characterized by a mean of $3$ and a standard deviation of $1.5$.

  • Objective: Determine the probability that the sample average is smaller than $2.3$.

  • Applied framework: Central Limit Theorem for averages, utilizing sample size $n = 36$, population mean $\mu = 3$, and standard deviation $\sigma = 1.5$.

• Single Value Percentile Problem:

  • Scenario: A single data value is evaluated from a normal distribution having a mean of $65$ and a standard deviation of $10$.

  • Objective: Identify the specific data value that marks the $40\text{th}$ percentile of this distribution.

  • Applied framework: Standard single-variable normal distribution analysis (CLT is not applied).

• Sample Sum Calculation Problem:

  • Scenario: A sample of $100$ values is randomly pulled from a binomial distribution $X \sim B(20, 0.5)$.

  • Objective: Determine the probability that the sample sum is between $1990$ and $2000$.

  • Applied framework: Central Limit Theorem for sums, using sample size $n = 100$ applied across the underlying binomial parameters.