Statistics Review: Confidence Intervals & Hypothesis Testing

Understanding Statistical Inference: Confidence Intervals and Hypothesis Testing

Focus on Logic, Not Memorization of Proofs

• The primary goal is to understand the logic behind statistical results, not to memorize complex proofs.
• Proofs are useful to understand once to grasp the underlying meaning of a result, such as knowing that $X^*$ is a random variable with a specific distribution and the logic for calculating a confidence interval from it.
• The emphasis is on the practical implications and understanding what the results mean, rather than the exact mathematical derivations.

Population Parameters and Sampling

• Population too big: Often, we deal with populations that are too large to measure every member.
• Unknown Parameters: We don't know the true parameters of these populations (e.g., mean ($\mu$), variance ($\sigma^2$), standard deviation ($\sigma$)).
• Comparison of Populations: We might want to compare parameters between two or more populations (e.g., $\mu<em>{x1}$ vs. $\mu</em>{x2}$ or ratio of variances).
• Sampling: To learn about population parameters, we take a sample, which is a set of observations.
• Random Event: The act of drawing a sample is a random event, which can be represented by random variables.
  • Example: Choosing $10$ students from MSU results in different sets of students each time.

Point Estimators

• Definition: A single value (a statistic derived from a sample) used to estimate an unknown population parameter.
• Formation: Sample data is 'put together' (e.g., summed, averaged) to estimate parameters.
  • Examples:
    • Sum: $\sum X_i$
    • Average (Sample Mean): $\bar{X} = \frac{1}{n} \sum{i=1}^{n} Xi
    • Sample Variance: $s^2 = \frac{\sum (X_i - \bar{X})^2}{n-1}$ (an unbiased estimator for population variance).
    • Alternative Sample Variance: $s<em>b^2 = \frac{\sum (X</em>i - \bar{X})^2}{n}$ (often called the maximum likelihood estimator, but it is a biased estimator for population variance).
• Properties of Estimators:
  • Unbiasedness: An estimator $\hat{\theta}$ is unbiased if its expected value equals the true population parameter $\theta$ (i.e., $E[\hat{\theta}] = \theta$).
    • Example: $E[\bar{X}] = \mu_X$ (sample mean is an unbiased estimator of population mean).
    • Example: $E[s^2] = \sigma^2$ (sample variance with $n-1$ in denominator is an unbiased estimator of population variance).
• Limitations of Point Estimators: Even if unbiased, a single point estimate doesn't convey the quality or accuracy of the estimation. Increasing sample size ($n$) generally improves quality, but we need more precise measures.

Interval Estimators (Confidence Intervals)

• Purpose: To provide a range of values (an interval) within which the true population parameter is likely to lie, along with a specified level of confidence.
• Formulation: Finding two estimators (a lower bound $\hat{\theta}<em>1$ and an upper bound $\hat{\theta}</em>2$) such that the probability of the true parameter $\theta$ being between them is $1-\alpha$ (e.g., $P(\hat{\theta}<em>1 \le \theta \le \hat{\theta}</em>2) = 1-\alpha$).
  • Example: The mean height of MSU students is between $5.7$ and $5.8$ feet with $99.95\%$ confidence.
  • Interpretation: If we repeat the sampling and interval calculation many times, $1-\alpha$ of these intervals would contain the true population parameter. (e.g., if $1-\alpha = 0.9995$, then in $10,000$ universities, $9995$ of the calculated intervals would contain the actual mean height).
• Derivation for Population Mean ($\mu$) with Known Population Standard Deviation ($\sigma$):
  • Assumption: For sufficiently large sample sizes ($n \geq 30$), the sample mean $\bar{X}$ is approximately normally distributed: $\bar{X} \sim N(\mu, \frac{\sigma^2}{n})$.
  • Standardization: Convert $\bar{X}$ to a standard normal variable (Z-score): $Z = \frac{\bar{X} - \mu}{\sigma/\sqrt{n}}$.
  • Probability Statement: From the Z-distribution table, we find critical values $\pm Z<em>{\alpha/2}$ such that: $P(-Z</em>{\alpha/2} \le Z \le Z_{\alpha/2}) = 1-\alpha$
  • Rearranging to find Confidence Interval for $\mu$:
    $P(\bar{X} - Z<em>{\alpha/2}\frac{\sigma}{\sqrt{n}} \le \mu \le \bar{X} + Z</em>{\alpha/2}\frac{\sigma}{\sqrt{n}}) = 1-\alpha$
  • The lower bound is $\hat{\theta}<em>1 = \bar{X} - Z</em>{\alpha/2}\frac{\sigma}{\sqrt{n}}$ and the upper bound is $\hat{\theta}<em>2 = \bar{X} + Z</em>{\alpha/2}\frac{\sigma}{\sqrt{n}}$.
• Derivation for Population Mean ($\mu$) with Unknown Population Standard Deviation ($\sigma$) or Small Sample Size ($n < 30$):
  • Use the t-distribution instead of the Z-distribution.
  • Replace population standard deviation $\sigma$ with sample standard deviation $s$: $T = \frac{\bar{X} - \mu}{s/\sqrt{n}}$ with $n-1$ degrees of freedom.
  • The process for finding the interval is otherwise similar.
• Derivation for Population Variance ($\sigma^2$):
  • Uses the Chi-squared ($\chi^2$) distribution.
  • Test Statistic: $\frac{(n-1)s^2}{\sigma^2} \sim \chi^2_{n-1}$.
  • Probability Statement: Find critical values $\chi^2<em>{\alpha/2}$ and $\chi^2</em>{1-\alpha/2}$ from the Chi-squared table such that:
    $P(\chi^2<em>{\alpha/2} \le \frac{(n-1)s^2}{\sigma^2} \le \chi^2</em>{1-\alpha/2}) = 1-\alpha$
    * (Note: The speaker clarifies that for the upper bound of the interval for $\chi^2$, one typically uses $\chi^2<em>{1-\alpha/2}$, which corresponds to the smaller critical value on the left tail, and $\chi^2</em>{\alpha/2}$ for the right tail, since the inequality is inverted when solving for $\sigma^2$)
  • Rearranging to find Confidence Interval for $\sigma^2$:
    $P(\frac{(n-1)s^2}{\chi^2<em>{1-\alpha/2}} \le \sigma^2 \le \frac{(n-1)s^2}{\chi^2</em>{\alpha/2}}) = 1-\alpha$
• Controlling Confidence Interval Properties:
  • Width vs. Confidence: To increase confidence ($1-\alpha$) while keeping sample size ($n$) fixed, the interval must become wider.
  • Width vs. Sample Size: To keep the interval width fixed while increasing confidence, or to narrow the interval for fixed confidence, the sample size ($n$) must be increased.
  • This trade-off helps in deciding optimal sample size based on desired precision and confidence.
• Other Parameters: Similar procedures exist for other parameters like proportions ($p<em>1 - p</em>2$) or ratios of variances ($\sigma1^2 / \sigma2^2 which uses the F-distribution, as seen in Fisher's technique).

Hypothesis Testing

• Core Idea: To check if a claim or belief, formulated as a hypothesis, is supported by available data and evidence.
  • Example: A belief that MSU students are taller than Missouri students.
• Formulating Hypotheses:
  • Null Hypothesis ($H_0$): A statement about a population parameter that is assumed to be true until evidence suggests otherwise. It typically includes an equality.
    • Example: The mean height of MSU students is $5.6$ feet ($H_0: \mu = 5.6$).
  • Alternative Hypothesis ($H<em>a$ or $H</em>1$): A statement that contradicts the null hypothesis. It can be:
    • Two-sided: The parameter is not equal to a specific value (e.g., $H_a: \mu \neq 5.6$).
    • One-sided: The parameter is greater than or less than a specific value (e.g., $H<em>a: \mu > 5.6$ or Ha: \mu < 5.6).
  • The null and alternative hypotheses should collectively cover all possibilities in the parametric space.
• Logic of Hypothesis Testing:
  • Assumption: Assume the null hypothesis ($H_0$) is true.
  • Sampling and Test Statistic: Take a sample and calculate a test statistic (e.g., $\bar{X}$).
  • Probability under $H<em>0$: Determine the probability of observing a test statistic as extreme as, or more extreme than, the one obtained, assuming $H</em>0$ is true.
  • Decision: If this probability is very low (below a predetermined significance level $\alpha$), then it casts doubt on the initial assumption that $H<em>0$ is true, and we reject $H</em>0$. Otherwise, we fail to reject $H_0$.
• Critical Region (Rejection Region):
  • Significance Level ($\alpha$): A chosen threshold (e.g., $0.05$ or $5\%$). It represents the maximum probability of making a Type I error.
  • Critical Values: Based on $\alpha$ and the distribution of the test statistic (assuming $H_0$ is true), we find critical values that define the boundaries of the critical region.
  • Decision Rule: If the calculated test statistic falls within the critical region, we reject $H_0$.
  • Example for Mean ($\mu$) with Known $\sigma$ (Two-sided test $H<em>0: \mu = \mu</em>0$ vs. $H<em>a: \mu \neq \mu</em>0$):
    • Test statistic: $Z = \frac{\bar{X} - \mu_0}{\sigma/\sqrt{n}}$.
    • Critical Region: Reject $H<em>0$ if $Z < -Z</em>{\alpha/2}$ or Z > Z_{\alpha/2}.
• Conclusion Statement:
  • If the test statistic is in the critical region: