Statistics and Data Analysis

1. Estimation of the Population Mean

a) Key Concepts and Definitions

• Estimator: A function of a sample used to estimate an unknown population parameter (e.g., population mean $m_Y$). It is a random variable because it depends on the random sample.
• Estimate: The numerical value obtained by applying an estimator to a specific sample.
• Sampling distribution: The probability distribution of an estimator over repeated samples.

b) Desirable Properties of Estimators

An estimator should be unbiased, consistent, and efficient to be considered good.

c) The Sample Mean $\bar{Y}$ as an Estimator of $m_Y$

• Given i.i.d. observations $Y_1, \dots, Y_n$ from the population, the sample mean is:

  $$\bar{Y} = \frac{1}{n} \sum_{i=1}^n Y_i$$

• Unbiasedness: $E(\bar{Y}) = m_Y$; thus, $\bar{Y}$ is unbiased.

• Consistency: By the Law of Large Numbers, $\bar{Y} \xrightarrow{p} m_Y$.

• Variance:

  $$Var(\bar{Y}) = \frac{\sigma_Y^2}{n}$$

  where $\sigma_Y^2$ is the population variance.

• Efficiency: Among all unbiased linear estimators (weighted averages of $Y_i$), $\bar{Y}$ has the smallest variance and is thus the Best Linear Unbiased Estimator (BLUE).

d) Comparison with Other Estimators

• Using only the first observation $Y_1$ as an estimator:

  • Unbiased: $E(Y_1) = m_Y$
  • Variance: $Var(Y_1) = \sigma_Y^2$
  • Since $Var(\bar{Y}) = \sigma_Y^2 / n < \sigma_Y^2$ for $n \geq 2$, $\bar{Y}$ is more efficient than $Y_1$.

• Weighted averages with unequal weights (e.g., alternating weights $1/2$ and $3/2$) have larger variance than $\bar{Y}$ and are less efficient.

e) Least Squares Estimator Interpretation

• The sample mean $\bar{Y}$ minimizes the sum of squared deviations from the data:

  $$\min_m \sum_{i=1}^n (Y_i - m)^2$$

• This makes $\bar{Y}$ the least squares estimator of $m_Y$.

f) Importance of Random Sampling

• The i.i.d. assumption (independent and identically distributed) is crucial for unbiasedness and consistency.
• Non-random sampling can introduce bias. For example, surveying only unemployed people in parks at work hours overrepresents unemployed individuals, biasing the estimate.
• Proper random sampling ensures the sample is representative of the population.

Key takeaway: The sample mean $\bar{Y}$ is the unbiased, consistent, and most efficient linear estimator of the population mean $m_Y$ when samples are drawn randomly and independently.

1. Hypothesis Tests Concerning the Population Mean

a) Null and Alternative Hypotheses

• Null hypothesis ($H_0$): The population mean equals a specified value $m_{Y,0}$.

  $$H_0: E(Y) = m_{Y,0}$$

• Alternative hypothesis ($H_1$): The population mean differs from $m_{Y,0}$.

  • Two-sided alternative:

    $$H_1: E(Y) \neq m_{Y,0}$$

  • One-sided alternatives are also possible but discussed separately.

• Hypothesis testing decides whether to reject $H_0$ or fail to reject $H_0$ based on sample data.

b) The $p$-Value

• The $p$-value is the probability, assuming $H_0$ is true, of observing a sample mean at least as extreme as the one actually observed.

• Formally, if $Y_{\text{act}}$ is the observed sample mean,

  $$p\text{-value} = \Pr_{H_0}\left(|Y - m_{Y,0}| \geq |Y_{\text{act}} - m_{Y,0}|\right)$$

• Interpretation:

  • Small $p$-value (e.g., 0.1%) → strong evidence against $H_0$, reject $H_0$.
  • Large $p$-value (e.g., 40%) → weak evidence against $H_0$, fail to reject $H_0$.

• To compute $p$-values, the sampling distribution of $Y$ under $H_0$ is needed.

c) Sampling Distribution of $Y$ and Large-Sample Approximation

• For large sample size $n$, by the Central Limit Theorem, $Y$ is approximately normally distributed under $H_0$:

  $$Y \sim N\left(m_{Y,0}, \frac{s_Y^2}{n}\right)$$

  where $s_Y^2$ is the population variance of $Y$.

• The standardized variable:

  $$Z = \frac{Y - m_{Y,0}}{s_Y / \sqrt{n}} \sim N(0,1)$$

d) Calculating the $p$-Value When $s_Y$ is Known

• If population variance $s_Y^2$ is known, the $p$-value for a two-sided test is:

  $$p\text{-value} = 2 \Phi\left(-\left|\frac{Y_{\text{act}} - m_{Y,0}}{s_Y / \sqrt{n}}\right|\right)$$

  where $\Phi$ is the standard normal CDF.

e) Sample Variance, Sample Standard Deviation, and Standard Error

| Standard error of $\bar{Y}$ | $$ SE(\bar{Y}) = \frac{s_Y}{\sqrt{n}} $$ | Estimator of the standard deviation of the sampling distribution of $\bar{Y}$. | - The sample variance is **consistent**: $s_Y^2 \xrightarrow{p} s_Y^2$ as $n \to \infty$. - For Bernoulli variables with success probability $p$, variance simplifies to $p(1-p)/n$ and

SE(\bar{Y}) = \sqrt{\frac{\bar{Y}(1-\bar{Y})}{n}}

#### f) Calculating the $p$-Value When $s_Y$ is Unknown - Replace $s_Y$ by $s_Y$ estimated from the sample:

p\text{-value} = 2 \Phi\left(-\left|\frac{Y_{\text{act}} - m_{Y,0}}{SE(\bar{Y})}\right|\right)

- This uses the **standard error** $SE(\bar{Y}) = s_Y / \sqrt{n}$. #### g) The $t$-Statistic - Defined as the standardized sample mean using the estimated standard error:

t = \frac{\bar{Y} - m_{Y,0}}{SE(\bar{Y})}

- The $t$-statistic is a **test statistic** used to perform hypothesis tests about the population mean. - For large $n$, by the Central Limit Theorem:

t \approx N(0,1) \quad \text{under } H_0

> **Key point:** The $p$-value quantifies the evidence against $H_0$ by measuring how extreme the observed sample mean is relative to its sampling distribution under $H_0$. The $t$-statistic standardizes this difference using the estimated standard error.

1. Standard Error of the Sample Mean

• The standard error (SE) of the sample mean $Y$ is denoted $SE(Y)$ or $s_Y / \sqrt{n}$.
• When $Y_1, \ldots, Y_n$ are i.i.d., $$SE(Y) = \frac{s_Y}{\sqrt{n}}$$ where $s_Y$ is the sample standard deviation.

2. Hypothesis Testing for the Population Mean

• The $t$-statistic for testing $H_0: \mu_Y = \mu_{Y,0}$ is $$t_{act} = \frac{Y_{act} - \mu_{Y,0}}{SE(Y)}.$$
• For large $n$, $t_{act}$ approximately follows a standard normal distribution $N(0,1)$ under $H_0$.
• The two-sided $p$-value is $$p\text{-value} = 2 \Phi(-|t_{act}|),$$ where $\Phi$ is the standard normal CDF.

3. Fixed Significance Level Testing

• Choose a significance level $\alpha$ (commonly 5%).
• Reject $H_0$ if $|t_{act}| > z_{\alpha/2}$, where $z_{\alpha/2}$ is the critical value from $N(0,1)$.
• For $\alpha=5\%$, $z_{0.025} = 1.96$.
• This controls the Type I error rate (false positive rate) at $\alpha$.

To reject $H_0$ at 5% significance level, require:

$$|t_{act}| > 1.96.$$

4. One-Sided Alternatives

• Alternative hypothesis can be one-sided, e.g. $$H_1: \mu_Y > \mu_{Y,0}.$$
• The $p$-value for one-sided test is $$p\text{-value} = 1 - \Phi(t_{act}).$$
• Reject $H_0$ if $t_{act} > z_\alpha$ (e.g., $z_{0.05} = 1.64$ for 5% level).
• For $H_1: \mu_Y < \mu_{Y,0}$, reject if $t_{act} < -z_\alpha$.

5. Confidence Intervals for the Population Mean

• A confidence interval (CI) is a range of values that contains the true population mean $\mu_Y$ with a specified probability (confidence level).

• A two-sided 95% CI for $\mu_Y$ is

  $$\left[Y - 1.96 \times SE(Y), \quad Y + 1.96 \times SE(Y)\right].$$

• More generally, for large $n$:

  Confidence Level	Critical Value ($z_{\alpha/2}$)	Confidence Interval
  90%	1.64	$Y \pm 1.64 \times SE(Y)$
  95%	1.96	$Y \pm 1.96 \times SE(Y)$
  99%	2.58	$Y \pm 2.58 \times SE(Y)$

• The coverage probability is the probability that the CI contains the true mean, computed over repeated samples.

6. Interpretation and Construction of Confidence Intervals

• The CI can be seen as the set of all $\mu_{Y,0}$ values not rejected by a two-sided hypothesis test at the chosen significance level.
• For example, if $Y=22.64$ and $SE(Y)=1.28$, the 95% CI is $$22.64 \pm 1.96 \times 1.28 = [20.13, 25.15].$$
• This means we are 95% confident that the true mean lies within this interval.

Key takeaway:
A 95% confidence interval contains the true population mean in 95% of all possible random samples.

1. Hypothesis Tests for the Difference Between Two Means

• Goal: Test if the difference between two population means, $m_m$ (men) and $m_w$ (women), equals a specified value $d_0$.

• Hypotheses:

  $$H_0: m_m - m_w = d_0 \quad \text{vs.} \quad H_1: m_m - m_w \neq d_0$$

  The special case $d_0=0$ tests equality of means.

• Estimator: From samples of sizes $n_m$ and $n_w$, sample means $Y_m$ and $Y_w$ estimate $m_m$ and $m_w$. The estimator for the difference is:

  $$Y_m - Y_w$$

• Sampling distribution: Assuming independence and approximate normality (by CLT),

  $$Y_m - Y_w \sim N\left(m_m - m_w, \frac{s_m^2}{n_m} + \frac{s_w^2}{n_w}\right)$$

  where $s_m^2$ and $s_w^2$ are population variances (usually unknown, replaced by sample variances).

• Standard error:

  $$SE(Y_m - Y_w) = \sqrt{\frac{s_m^2}{n_m} + \frac{s_w^2}{n_w}}$$

• Test statistic:

  $$t = \frac{(Y_m - Y_w) - d_0}{SE(Y_m - Y_w)}$$

  For large $n_m$ and $n_w$, $t$ approximately follows a standard normal distribution under $H_0$.

• Decision rule:

  • Two-sided test at 5% level: reject $H_0$ if $|t| > 1.96$.
  • One-sided test (e.g., $H_1: m_m - m_w > d_0$): reject if $t > 1.64$.

• P-value: Computed using the standard normal distribution as in single mean tests.

2. Confidence Intervals for the Difference Between Two Means

• 95% confidence interval for $d = m_m - m_w$: $$(Y_m - Y_w) \pm 1.96 \times SE(Y_m - Y_w)$$
• This interval contains all values $d_0$ not rejected by the two-sided test at the 5% level.

3. Differences-of-Means Estimation of Causal Effects Using Experimental Data

• Randomized controlled experiment: Subjects randomly assigned to treatment ($X=x$) or control ($X=0$) groups.

• Causal effect (treatment effect):

  $$\text{Causal effect} = E(Y|X=x) - E(Y|X=0)$$

  For binary treatment ($X=0$ control, $X=1$ treatment):

  $$\text{Treatment effect} = E(Y|X=1) - E(Y|X=0)$$

• Estimation: Difference in sample means between treatment and control groups estimates the causal effect.

• Testing: Use the $t$-statistic for difference of means (Equation above).

• Confidence interval: Use the confidence interval for difference of means as a confidence interval for the causal effect.

• Note: Randomized experiments provide strong causal inference but may be costly or unethical in some fields; quasi-experiments can mimic randomization.

4. Using the $t$-Statistic When the Sample Size Is Small

• For small samples, the standard normal approximation for the $t$-statistic may be poor.

• If the population is normally distributed, the exact distribution of the $t$-statistic for a single mean is the Student $t$ distribution with $n-1$ degrees of freedom.

• Critical values for hypothesis tests and confidence intervals should be taken from the Student $t$ distribution in small samples.

5. Summary Table: Comparing Two Means

Key point: The difference in sample means is a natural estimator for the difference between population means and for causal effects in randomized experiments; hypothesis testing and confidence intervals rely on the $t$-statistic with appropriate distributional assumptions.

1. Differences-of-Means Estimation of Causal Effects Using Experimental Data

a) Estimating Treatment Effects by Differences of Means

• Causal effect of a treatment is estimated by the difference in average outcomes between the treatment and control groups.

• Let $Y_i(1)$ be the potential outcome if unit $i$ receives treatment, and $Y_i(0)$ if not.

• The average treatment effect (ATE) is $E[Y_i(1) - Y_i(0)]$.

• In randomized experiments, the difference in sample means between treated and control groups is an unbiased estimator of the ATE:

  $$\hat{\tau} = \bar{Y}_1 - \bar{Y}_0$$

  where $\bar{Y}_1$ and $\bar{Y}_0$ are sample means of treated and control groups respectively.

b) Hypothesis Testing Using the $t$-Statistic

• To test hypotheses about means, the $t$-statistic is used:

  $$t = \frac{\bar{Y} - \mu_0}{SE(\bar{Y})}$$

  where $SE(\bar{Y})$ is the standard error of the sample mean.

• For small samples and normally distributed data, $t$ follows a Student $t$ distribution with $n-1$ degrees of freedom.

• For large samples, the $t$-statistic approximates a standard normal distribution.

c) $t$-Statistic for Differences of Means

• The $t$-statistic for testing difference between two means is:

  $$t = \frac{\bar{Y}_m - \bar{Y}_w}{SE(\bar{Y}_m - \bar{Y}_w)}$$

• The standard error allowing for unequal variances is:

  $$SE(\bar{Y}_m - \bar{Y}_w) = \sqrt{\frac{s_m^2}{n_m} + \frac{s_w^2}{n_w}}$$

• The $t$-statistic does not follow a Student $t$ distribution in general when variances differ.

• If variances are equal and sample sizes are equal or large, the pooled variance estimator can be used:

  $$s^2_{\text{pooled}} = \frac{1}{n_m + n_w - 2} \left( \sum_{i=1}^{n_m} (Y_i - \bar{Y}_m)^2 + \sum_{i=1}^{n_w} (Y_i - \bar{Y}_w)^2 \right)$$

• The pooled standard error is:

  $$SE_{\text{pooled}}(\bar{Y}_m - \bar{Y}_w) = s_{\text{pooled}} \sqrt{\frac{1}{n_m} + \frac{1}{n_w}}$$

• Under equal variances, the pooled $t$-statistic follows a Student $t$ distribution with $n_m + n_w - 2$ degrees of freedom.

• Caution: Use pooled variance only if variances are equal; otherwise, it is biased and invalid.

d) Practical Use of the Student $t$ Distribution

• Economic data rarely follow normal distributions exactly; however, for large samples, the normal approximation to the $t$-statistic distribution is valid.

• For differences in means, always use the unequal variance standard error and rely on the large-sample normal approximation.

• Differences between Student $t$ and normal distributions become negligible as sample size grows (e.g., for $n > 80$, difference in $p$-values is < 0.002).

e) Summary Table: $t$-Statistic for Means

The pooled $t$-statistic should be used only if there is strong evidence that population variances are equal; otherwise, use the unequal variance formula with large-sample normal approximation.

2. Application: Increasing Voter Turnout via Experimental Interventions

• Experiments using canvassing and telephone calls increased voter turnout significantly compared to control groups.

• Estimated effects were approximately 6.7% and 7.3% increases in participation.

• Statistical significance was assessed using differences-of-means estimators and appropriate $t$-tests.

• This illustrates how randomized experiments and differences-of-means estimation can quantify causal effects in social science.

3. Key Formulas Recap

• Difference-in-means estimator:

  $$\hat{\tau} = \bar{Y}_1 - \bar{Y}_0$$

• Standard error (unequal variances):

  $$SE(\hat{\tau}) = \sqrt{\frac{s_1^2}{n_1} + \frac{s_0^2}{n_0}}$$

• Pooled variance (equal variances):

  $$s^2_{\text{pooled}} = \frac{1}{n_1 + n_0 - 2} \left( \sum (Y_i - \bar{Y}_1)^2 + \sum (Y_i - \bar{Y}_0)^2 \right)$$

• Pooled standard error:

  $$SE_{\text{pooled}}(\hat{\tau}) = s_{\text{pooled}} \sqrt{\frac{1}{n_1} + \frac{1}{n_0}}$$

• $t$-statistic for difference of means:

  $$t = \frac{\hat{\tau}}{SE(\hat{\tau})}$$

Use differences-of-means estimators with appropriate standard errors and $t$-tests to estimate and infer causal effects from experimental data.

1. Definition of the t-Statistic

For a sample of size $n$ with sample mean $\bar{Y}$ and sample standard deviation $s$, testing the null hypothesis $H_0: \mu = \mu_0$ uses the statistic:

$$t = \frac{\bar{Y} - \mu_0}{s / \sqrt{n}}$$

This statistic follows a t-distribution with $n-1$ degrees of freedom (df).

2. Key Properties of the t-Distribution

3. When to Use the t-Statistic

• The population is normally distributed or approximately normal.
• The sample size $n$ is small (commonly $n < 30$).
• The population standard deviation $\sigma$ is unknown.
• The sample standard deviation $s$ is used as an estimate of $\sigma$.

4. Confidence Intervals Using the t-Distribution

A $100(1-\alpha)\%$ confidence interval for the population mean $\mu$ is:

$$\bar{Y} \pm t_{\alpha/2, n-1} \times \frac{s}{\sqrt{n}}$$

where $t_{\alpha/2, n-1}$ is the critical value from the t-distribution with $n-1$ degrees of freedom.

5. Hypothesis Testing Using the t-Statistic

• Null hypothesis: $H_0: \mu = \mu_0$
• Alternative hypothesis: $H_1: \mu \neq \mu_0$ (two-sided) or $H_1: \mu > \mu_0$ / $\mu < \mu_0$ (one-sided)
• Compute the t-statistic as above.
• Compare $|t|$ to the critical value $t_{\alpha/2, n-1}$ for two-sided tests or $t_{\alpha, n-1}$ for one-sided tests.
• Alternatively, compute the p-value as the probability of observing a t-statistic as extreme as the computed value under $H_0$.
• Reject $H_0$ if $|t| > t_{\alpha/2, n-1}$ or if the p-value is less than the significance level $\alpha$.

6. Degrees of Freedom and Their Importance

• The degrees of freedom reflect the number of independent pieces of information used to estimate the population variance.
• Using $n-1$ degrees of freedom corrects the bias in the estimation of the population variance from the sample.

7. Summary Table: z-Statistic vs t-Statistic

Key point: When the sample size is small and the population standard deviation is unknown, use the t-statistic with $n-1$ degrees of freedom for inference about the population mean.

8. Practical Notes

• The t-distribution is robust to moderate departures from normality, especially as $n$ increases.
• For very small samples, normality of the population is a critical assumption.
• Software and statistical tables provide critical values and p-values for the t-distribution.

This approach ensures valid inference on the population mean despite the increased uncertainty from estimating the standard deviation with a small sample.

1. Scatterplots

• A scatterplot is a graphical representation of the relationship between two quantitative variables.
• Each point represents an observation with coordinates $(x_i, y_i)$.
• Scatterplots help visualize patterns, trends, clusters, and outliers.

2. Sample Covariance

• Measures the direction and strength of linear association between two variables $X$ and $Y$ in a sample.

• Defined as:

  $$s_{XY} = \frac{1}{n-1} \sum_{i=1}^n (X_i - \bar{X})(Y_i - \bar{Y})$$

• Interpretation:

  • $s_{XY} > 0$: positive linear relationship.
  • $s_{XY} < 0$: negative linear relationship.
  • $s_{XY} = 0$: no linear relationship.

• Units depend on the units of $X$ and $Y$, so covariance alone is not standardized.

3. Sample Correlation

• The sample correlation coefficient $r$ standardizes covariance to a dimensionless measure between $-1$ and $1$.

• Defined as:

  $$r = \frac{s_{XY}}{s_X s_Y}$$

  where $s_X$ and $s_Y$ are the sample standard deviations of $X$ and $Y$.

• Properties:

  • $r \in [-1, 1]$.
  • $r = 1$: perfect positive linear relationship.
  • $r = -1$: perfect negative linear relationship.
  • $r = 0$: no linear relationship.

• Measures strength and direction of linear association.

4. Key Formulas Summary

5. Important Notes

• Covariance is sensitive to scale; correlation is scale-invariant.
• Both covariance and correlation measure linear relationships only.
• Scatterplots are essential for visualizing the relationship before computing covariance or correlation.
• The sample covariance is an unbiased estimator of the population covariance under i.i.d. assumptions.
• The sample correlation coefficient is a consistent estimator of the population correlation.

The sample correlation coefficient $r$ quantifies the strength and direction of the linear relationship between two variables, normalized to lie between $-1$ and $1$.