STAT 100 Lesson 9: Confidence Intervals for Proportions and Means

Introduction to Confidence Intervals and Inferential Statistics

• Lesson 9 in STAT 100 transitions towards the end of the course into the realm of inferential statistics.

• The primary goal is to make estimates about unknown population parameters when researchers do not have access to the entire population.

• This process involves taking a random sample and making an inference from it.

• Types of population parameters discussed include:

  • Proportions (e.g., percentage of a group).

  • Means (e.g., average measurements).

  • Difference of two means.

  • Difference of two proportions.

Parameters vs. Statistics

• It is vital to distinguish between values pertaining to the population versus values derived from a sample.

• Parameter: A value that describes the whole population.

  • Notation for proportions: $P$ (the true proportion).

  • Notation for means: $\mu$ (the true population mean).

• Statistic: A value calculated from a random sample.

  • Notation for proportions: $\hat{p}$ (pronounced "p-hat").

  • Notation for means: $\bar{x}$ (pronounced "x-bar").

Case Study 1: Veterans Health Care (Proportions)

• The Problem: Researchers are interested in the percentage of veterans dissatisfied with the health care system.

• Data: A random sample of $n = 400$ veterans is taken. 100 veterans report dissatisfaction.

• Parameter of Interest: The proportion of all veterans dissatisfied with the health care system.

• Statistic: $\hat{p} = \frac{100}{400} = 0.25$.

Case Study 2: Equatorial Diameter of Jupiter (Means)

• The Problem: Measuring the true equatorial diameter of Jupiter using independent measurement tools.

• Assumptions: The process is free of bias or represents a random sample.

• Data: A sample of size $n = 36$ is taken. The average of these values is $88,720\,\text{miles}$ with a standard deviation of $72\,\text{miles}$.

• Parameter of Interest: The actual equatorial diameter of Jupiter ($\mu$).

• Statistic: $\bar{x} = 88,720\,\text{miles}$. The sample standard deviation is $s = 72\,\text{miles}$.

The Nature of Random Samples and Distributions

• Random samples exhibit a specific order. If one takes random samples of the same size repeatedly from a population, the sample statistics will approximate a bell-shaped (Normal) distribution.

• Sampling Distribution of Proportions: Sentered right at the true population proportion ($P$).

• Sampling Distribution of Averages: Centered right at the true population mean ($\mu$).

• The Reality of Sampling: In practice, researchers only have one sample. For the veterans study, it is the sample where $\hat{p} = 0.25$. For Jupiter, it is the measurement where $\bar{x} = 88,720$. It is impossible to know exactly where a single sample falls on the distribution without knowing the population parameter.

The Normal Approximation and Conditions for Validity

• The sampling distribution follows a normal curve if specific conditions are met:

  • For Proportions:

    • $n \times p \geq 5$

    • $n \times (1 - p) \geq 5$

    • In the VA example: $400 \times 0.25 = 100$ and $400 \times 0.75 = 300$. Both are well above the threshold of 5.

  • For Means:

    • The sample size requirements depend on the skewness of the population distribution.

    • $n \geq 15$ if the distribution is not strongly skewed.

    • $n \geq 25$ if moderately skewed.

    • $n \geq 40$ if strongly skewed.

    • In the Jupiter example, $n = 36$, which is sufficient for moderate skewness.

Standard Error (SE)

• Since the population parameters ($P$ and $\sigma$) are unknown, we cannot calculate the exact standard deviation of the sampling distribution. Instead, we calculate the "Standard Error" using sample statistics.

• Relationship with Sample Size: As sample size ($n$) increases, the standard error decreases. This is an inverse relationship that increases the precision of the estimate.

• Formula for Proportions:

  • $SE_{\hat{p}} = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$

• Formula for Means:

  • $SE_{\bar{x}} = \frac{s}{\sqrt{n}}$

The General Form of a Confidence Interval

• A confidence interval recognizes that a sample statistic equals the population parameter plus some margin of error.

• General Formula: $\text{Sample Statistic} \pm \text{Margin of Error}$

• Margin of Error Calculation: $\text{Multiplier} \times \text{Standard Error}$

• The multiplier comes from the standard normal distribution ($Z$ distribution) with a center of 0 and a standard deviation of 1.

Z-Multipliers Table

• 80% Confidence: $Z^{*} = 1.282$

• 90% Confidence: $Z^{*} = 1.645$

• 95% Confidence: $Z^{*} = 1.96$ (often rounded to 2 in the Empirical Rule).

• 99% Confidence: $Z^{*} = 2.58$

Constructing and Interpreting Intervals (VA and Jupiter Examples)

VA Study (95% Confidence Interval)

• Statistic: $0.25$

• Multiplier: $1.96$

• Standard Error: $\sqrt{\frac{0.25(0.75)}{400}} = 0.0216$

• Calculation: $0.25 \pm 1.96 \times 0.0216$ results in the interval $[0.21, 0.29]$.

• Interpretation: We are 95% confident that the true percentage of veterans unsatisfied with the healthcare system is between 21% and 29%.

Jupiter Study (95% Confidence Interval)

• Statistic: $88,720$

• Multiplier: $1.96$

• Standard Error: $\frac{72}{\sqrt{36}} = 12$

• Calculation: $88,720 \pm 1.96 \times 12$ results in the interval $[88,696, 88,744]$.

• Interpretation: We are 95% confident the true equatorial diameter of Jupiter is between $88,696\,\text{miles}$ and $88,744\,\text{miles}$.

Advanced Interpretations and Logic

• The Meaning of Confidence: It does not mean there is a "95% chance" the true mean is there. It means if the sampling process were repeated many times, the calculated intervals would contain the true parameter 95% of the time.

• Evidence and Plausibility:

  • If a 95% interval for the VA study is $[0.21, 0.29]$, is it plausible the true proportion is less than 30%? Yes, because the entire interval is below 30%.

  • How to find the Margin of Error from an existing interval: Calculate the difference between the upper/lower bounds and the point estimate in the middle. For $[0.21, 0.29]$, the middle is $0.25$, and the margin of error is $0.04$.

• What Changes and What Stays Constant: If you take a new sample of 36 measurements for Jupiter, the true value remains constant. However, the sample statistic, the intervals, and the standard error may all change due to randomness.

Two-Sample Confidence Intervals (Difference in Proportions)

• The Problem: Comparison of tooth decay (untreated cavities) between sugary soda drinkers and non-soda drinkers.

• Group 1 (Soda): $n = 180$, $117$ have cavities ($\hat{p}_1 = 0.65$).

• Group 2 (Non-soda): $n = 220$, $88$ have cavities ($\hat{p}_2 = 0.40$).

Combining Standard Errors

• If the two samples are independent, the standard error for the difference is found by adding the squared individual standard errors then taking the square root:

  • $SE_{\text{diff}} = \sqrt{SE_1^2 + SE_2^2}$

• In this study, $SE_{diff} \approx 0.068$.

Calculating the 99% Interval for Difference

• Statistic: $\hat{p}_1 - \hat{p}_2 = 0.65 - 0.40 = 0.25$

• Multiplier: $2.58$

• Interval: $0.25 \pm 2.58 \times 0.068$

• Result: $[0.075, 0.425]$.

• Interpretation: We are 99% confident that the true difference in the proportion of teenagers with cavities between the two groups is between 0.075 and 0.425.

Significance Testing via Intervals

• Including Zero: If a difference interval includes zero (e.g., $[-0.11, 0.21]$), zero is a plausible difference, meaning we do not have enough evidence to claim a significant difference between the groups.

• Non-Overlapping Intervals: If individual intervals for two groups do not overlap (e.g., $[0.21, 0.28]$ vs $[0.31, 0.42]$), it strongly suggests a statistically significant difference exists.