Confidence Interval Estimation in Investment Analytics

Context: Week 4 focus on Confidence Interval Estimation within the framework of "Statistics for Investors," using the case study of TechVenture’s portfolio.
The Business Problem: TechVenture manages 50 portfolio companies with internal revenue data, but Limited Partners (LPs) require the average revenue of all Australian tech startups.
- The Reality: Thousands of startups exist across Australia, making the surveying of the entire population impossible due to high costs and time constraints.
- The Sample: The portfolio of 50 companies is the only available dataset.
- Key Question: Can the true population mean ( $\mu$ ) be estimated from a sample of $n = 50$ ?

Definition of Statistical Inference: The process of drawing conclusions about a population based on data from a sample.
Components of Inference:
- Population: All Australian tech startups; characterized by unknown revenues for thousands of companies and an unknown mean ( $\mu$ ).
- Sample: TechVenture's 50 portfolio companies; characterized by known revenues and a measured sample mean ( $\bar{x} = \$32,962$ ).
Point Estimates:
- Best Guess: The calculated sample mean ( $\bar{x}$ ) of $\$32,962$ serves as the point estimate for the population mean ( $\mu$ ).
- Limitations:
  - The population mean ( $\mu$ ) is almost certainly not exactly $\$32,962$ .
  - A different sample of 50 startups would produce a different $\bar{x}$ .
  - Point estimates provide no sense of uncertainty.

Definition: An interval estimate identifies a range of values that acknowledges uncertainty, providing a defensible range instead of false precision.
Comparison of Approaches:
- Point Estimate: "The mean is $\$32,962$ ."
- Interval Estimate: "We're 95% confident the mean is between $\$27,000$ and $\$39,000$ ."
Benefits of Interval Estimates:
- Honesty: Explicitly acknowledges that samples do not provide perfect population info.
- Precision Indicator: A wider interval indicates more uncertainty; a narrower interval indicates higher precision.
- Decision Support: Helps verify if the mean exceeds specific thresholds (e.g., if the CI is $[\$27k, \$39k]$ , one can be confident the mean is over $\$25k$ ).
- Sample Size Influence: Larger samples lead to narrower intervals and more confident estimates.

Sampling Distribution Thought Experiment:
1. Draw a random sample of 50 startups from the population.
2. Calculate the sample mean ( $\bar{x}$ ).
3. Repeat this process 10,000 times.
4. The resulting distribution of these 10,000 sample means is the "Sampling Distribution of the Mean."
Central Limit Theorem (CLT) Core Idea: For sufficiently large samples ( $n \geq 30$ ), the sampling distribution of $\bar{x}$ is approximately normal, regardless of the distribution of the underlying population.
Properties of the Sampling Distribution:
- Mean: Equal to the population mean ( $\mu$ ).
- Standard Deviation: Equal to $\frac{\sigma}{\sqrt{n}}$ , also known as the Standard Error ( $SE$ ).
Application to TechVenture:
- Sample size ( $n = 50$ ) exceeds the $n \geq 30$ threshold.
- Therefore, $\bar{x}$ follows an approximately normal distribution, even though individual startup revenues are right-skewed.

Definition: The Standard Error is the standard deviation of the sampling distribution, measuring how much sample means typically vary from the true population mean.
Formula: $SE = \frac{\sigma}{\sqrt{n}}$ (Since population $\sigma$ is rarely known, the sample standard deviation $s$ is substituted).
TechVenture Calculation:
1. Given: Sample standard deviation ( $s$ ) = $\$34,196.46$ ; Sample size ( $n$ ) = $50$ .
2. Formula Implementation: SE = \frac{\34,196.46}{\sqrt{50}} = \frac{\34,196.46}{7.071} .
3. Result: $SE = \$4,836.11$ .
Interpretation: If many samples of 50 startups were taken, the sample means would typically vary by approximately $\$4,836$ from the true population mean.
Takeaway: Larger $n$ results in a smaller $SE$ , leading to more precise estimates and narrower confidence intervals.

General Form: $\bar{x} \pm \text{Margin of Error}$ .
Full Breakdown: $\bar{x} \pm \text{Critical Value} \times \frac{s}{\sqrt{n}}$ .
Determining the Critical Value: Based on the Confidence Level and Distribution Type.
- $z$ -distribution: Used when population standard deviation ( $\sigma$ ) is known (Standard normal fixed shape).
- $t$ -distribution: Used when $\sigma$ is unknown (the realistic case). Shape depends on Degrees of Freedom ( $df = n - 1$ ). It has "fatter tails" to account for extra uncertainty about $\sigma$ .
Common Critical Values (z):
- 90% Confidence: $z = 1.645$
- 95% Confidence: $z = 1.96$
- 99% Confidence: $z = 2.576$
Excel Implementation for TechVenture ( $n=50$ , $df=49$ ):
- $z$ -distribution (95% CI): =NORM.S.INV(0.975) \rightarrow 1.96.
- $t$ -distribution (95% CI): =T.INV(0.975, 49) \rightarrow 2.01 or =T.INV.2T(0.05, 49) \rightarrow 2.01.
- Logic for 0.975: 95% confidence implies 5% in the tails, split as 2.5% per side. The critical value is pinpointed where 97.5% of the distribution lies to the left.

Definition of "95% Confident": If the sampling process were repeated many times and a 95% CI was calculated each time, approximately 95% of those resulting intervals would contain the true population mean ( $\mu$ ).
Common Misconception: It does NOT mean there is a 95% chance that $\mu$ falls in one specific interval. The population mean is fixed; the interval is the variable that changes from sample to sample.

Wider Intervals (Lower Precision/Higher Uncertainty):
- Higher confidence level (e.g., 99% vs 95%).
- Smaller sample size ( $n$ ).
- Higher variability in data (larger sample standard deviation $s$ ).
Narrower Intervals (Higher Precision):
- Lower confidence level (e.g., 90% vs 95%).
- Larger sample size ( $n$ ) — the easiest lever to control.
- Lower variability in data (smaller $s$ ).
The Trade-off: Maximum confidence and maximum precision cannot coexist; it is a fundamental design choice.

Inputs: $n = 50$ , $\bar{x} = \$32,962.25$ , $s = \$34,196.46$ .
90% CI:
- Lower Bound: $\$24,854.26$
- Upper Bound: $\$41,070.23$
- Width: $\$16,215.97$
95% CI:
- Lower Bound: $\$23,243.72$
- Upper Bound: $\$42,680.77$
- Width: $\$19,437.05$
99% CI:
- Lower Bound: $\$20,001.71$
- Upper Bound: $\$45,922.79$
- Width: $\$25,921.08$
Observation: The 99% CI is nearly $1.6 \times$ wider than the 90% CI.

Key Discussion Questions for TechVenture:
- What is the interval estimate for average startup revenue?
- Can we be 90% confident the true mean is above $\$25,000$ ?
- Can we be 90% confident the true mean is above $\$45,000$ ?
- How does increasing $n$ sharpen the interval?