Lecture 25 - Inferential Statistics: Extensive Guide to Confidence Intervals for Proportions and Means

The Paradigm of Inferential Statistics

• Transition from Descriptive to Inferential Statistics:

  • Descriptive statistics involve collecting data and computing basic measures such as means ($\bar{x}$), proportions ($\hat{p}$), and standard deviations ($s$).

  • Inferential statistics shift the focus to using these sample statistics to make authoritative statements about a population parameter.

• Two Primary Methods of Statistical Inference:

  1. Hypothesis Testing: Evaluates the plausibility of a single hypothesized value of a parameter. This is often called point estimation because it focuses on a specific point.

  2. Confidence Intervals: Calculates a range of plausible values that the population parameter might take. This is referred to as interval estimation.

• Key Population Parameters:

  • The two primary parameters of interest are the population mean ($\mu$) and the population proportion ($\pi$ or $p$).

  • Distinction between Statistics and Parameters: It is critical to distinguish between sample statistics (e.g., sample proportion) and population parameters. The purpose of this study is to use sample statistics to estimate population parameters.

Confidence Intervals for Population Proportion: A Comprehensive Recap

• The General Formula for Proportion CI:

  • $\hat{p} \pm z^* \times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

  • Components Defined:

    • Sample Estimate ($\hat{p}$): The proportion observed in the sample data.

    • Multiplier ($z^*$): A value determined by the desired confidence level, sourced from the standard normal (z) table.

    • Standard Error ($\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$): Measures the variability of the sample proportion across different samples.

    • Margin of Error: The product of the multiplier and the standard error ($z^* \times \text{Standard Error}$).

• Conservative Margin of Error:

  • Derived from lectures 6 and 7, this is calculated as $\frac{1}{\sqrt{n}}$.

• Common Multipliers ($z^*$) for the Normal Distribution:

  • 90% Confidence: $z^* = 1.645$

  • 95% Confidence: $z^* = 1.96$

• Interpretation of Confidence:

  • In frequentist theory, being "95% confident" means that if the sampling process were repeated 100 times, approximately 95 of those computed intervals would capture the true population proportion.

  • It is an objective reality based on long-run sampling, not a subjective feeling.

Theoretical Assumptions for Proportion Confidence Intervals

• 1. Representativeness: The sample must be representative of the population of interest. Using non-probability sampling methods generally results in invalid ("nonsense") data.

• 2. Independence: Samples must be collected independently. Probability-based (random) sampling usually satisfies this requirement by default.

• 3. Large Sample Size Conditions: The sample size $n$ must be large enough to satisfy:

  • n \times \hat{p} > 10

  • n \times (1 - \hat{p}) > 10

• Example Case (Alien Survey Data):

  • $n = 1003$, $\hat{p} = 0.56$

  • Check 1: $1003 \times 0.56 \approx 561.68$ (which is > 10).

  • Check 2: $1003 \times 0.44 \approx 441.32$ (which is > 10).

  • Conclusion: The assumption for sample size is met.

Confidence Intervals for Population Mean

• Structure of the Formula:

  • The structure is identical to that of proportions: $\text{Sample Statistic} \pm \text{Margin of Error}$.

  • Formula: $\bar{x} \pm t^* \times \frac{s}{\sqrt{n}}$

• Components Defined:

  • Sample Mean ($\bar{x}$): The statistic used to estimate the population mean ($\mu$).

  • Sample Standard Deviation ($s$): Used to calculate the standard error.

  • Sample Size ($n$): Number of observations in the sample.

  • Standard Error for Means: $\frac{s}{\sqrt{n}}$.

  • t-Multiplier ($t^*$): Obtained from Student's t-distribution rather than the normal (z) distribution.

The Student’s t-Distribution

• Characteristics:

  • Bell-shaped and centered at zero.

  • More spread out than the normal distribution with "wider tails" to account for increased uncertainty when using sample standard deviation.

• Degrees of Freedom ($df$):

  • Determined by $df = n - 1$.

  • The multiplier depends on both the degrees of freedom and the confidence level.

• Finding the t-Multiplier:

  • Using a t-table, align the $df$ (rows) with the confidence level (columns).

  • Example: For a 99% CI with 15 degrees of freedom ($df=15$), the multiplier is $2.95$.

  • Relationship to Normal Distribution: As $df \rightarrow \infty$, the t-distribution becomes narrower and effectively identical to the standard normal distribution. For infinite degrees of freedom, a 90% level yields a multiplier of $1.645$ and 95% yields $1.96$.

Practical Application: Human Body Temperature

• Background: An observational study from the Journal of Statistical Education tested if average human body temperature is actually $98.6\,^{\circ}\text{F}$.

• Sample Data:

  • $n = 30$

  • $\bar{x} = 98.25\,^{\circ}\text{F}$

  • $s = 0.73\,^{\circ}\text{F}$

• Calculation Walkthrough (95% CI):

  1. Find df: $30 - 1 = 29$.

  2. Find Multiplier ($t^*$): From the table, for $df = 29$ at 95% confidence, $t^* = 2.05$.

  3. Standard Error: $\frac{0.73}{\sqrt{30}} \approx 0.133279$.

  4. Margin of Error: $2.05 \times 0.133279 \approx 0.2732$.

  5. Lower Bound: $98.25 - 0.2732 = 97.9768$

  6. Upper Bound: $98.25 + 0.2732 = 98.5232$

  • Resulting Interval: $[97.97, 98.52]$

• Interpretation: We are 95% confident that the population average of human body temperatures is between $97.97\,^{\circ}\text{F}$ and $98.52\,^{\circ}\text{F}$.

• Comparison to Excel: Excel is more precise than the table. While we used $t^* = 2.05$, Excel uses $2.0452$, leading to very slight differences in limits.

Relationship Between Hypothesis Tests and Confidence Intervals

• Consistency: The two methods are mathematically related and consistent with each other.

  • In the body temperature example, the null hypothesis ($H_0 = 98.6\,^{\circ}\text{F}$) was rejected.

  • Correspondingly, the value $98.6$ falls outside the 95% confidence interval ($97.97$ to $98.52$).

  • Direct Linkage: A confidence level of $100 \times (1 - \alpha)\%$ corresponds directly to a significance level of $\alpha$. For instance, a 95% CI ($\alpha = 0.05$) will exclude any null hypothesis value that would be rejected by a two-tailed test at the $5\%$ level.

  • Proportion Caveat: This strict consistency holds perfectly for means. For proportions, the standard error formulas slightly differ (Hypothesis tests use the hypothesized $p_0$, while CI uses sample $\hat{p}$), but results remain generally consistent.

Assumptions for Confidence Intervals of Means

• 1. Representativeness: Sample must be representative of the population.

• 2. Independence: Data points must be independent (random sampling).

• 3. Large Sample Size:

  • General rule: $n \ge 30$.

  • If n < 30: You must either assume the population is normally distributed or verify that the sample data is fairly unimodal and symmetric without extreme outliers.

Questions & Discussion

• Q: Do we use the sample proportion or population proportion in the CI?

  • Response: We use the sample proportion ($\hat{p}$) because the population proportion is the unknown parameter we are trying to estimate. It’s a subtle but vital distinction.

• Q: Should we calculate the body temperature CI together as a class or individually?

  • Response: The class voted to calculate it together. The resulting range was approximately $97.97$ to $98.52$.

• Q: What is the correct way to start the interpretation?

  • Response: Always start with "We are [X]% confident that…". Ensure you mention the "population average" or "population mean" and define the context (e.g., body temperatures). Avoid terms like "sure" or "likely."