CH5 - Confidence Intervals

Research Question

When working with confidence intervals, we start with a Research Question:
Taking a sample and finding the sample statistic, what can we estimate about the population? To answer this, we calculate a confidence interval or conduct a Hypothesis Test.

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5.1 - Point Estimates and Confidence Intervals

Point estimates are single values calculated from sample data to estimate population parameters. They serve as our “best guess” of the true population value.

Point Estimates

For example:

• Use the sample mean $\bar{x}$ to estimate the population mean $\mu$.
• Use the sample proportion $\hat{p}$ to estimate the population proportion $p$.

Point estimates give us a starting point, but they lack information about how much error may exist in the estimate. That’s where confidence intervals come in.

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5.2 - Confidence Intervals for Proportions

Traditional Method Only

In this class, we focus solely on the traditional method for constructing confidence intervals, ignoring other approaches like bootstrapping.

Confidence Interval Example - Presidential Approval Rating

Polls often present approval ratings in the format:

$$45\%\pm 3\%$$

This is equivalent to the formula:

$$\hat{p}\pm E$$

Where:

• $\hat{p}$ = sample proportion
• $E$ = margin of error

Constructing a Confidence Interval

1. Start with a Random Sample from a Sampling Distribution. Assume the sample proportion $\hat{p}$ is approximately normal:

   $$\hat{p}N(p,\sqrt{\frac{p(1-p)}{n}})$$

   Where:

   • $p$ = population proportion
   • $n$ = sample size

2. Calculate the standard error (SE):

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

   For confidence intervals, $\sigma \approx SE(\hat{p})$.

3. Use the formula for a confidence interval:

   $$\hat{p}\pm z\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$

   Where $z$ is the critical value for the desired confidence level.

Confidence Levels and Critical Values

Common critical values for the $z$-score:

• 95% Confidence Level: $$C=95\%\to \frac{\alpha }{2}=0.025\to z_{.025}=|invNorm(0.025)|=1.96$$
• 90% Confidence Level: $$C=90\%\to \frac{\alpha }{2}=0.05\to z_{.05}=|invNorm(0.05)|=1.645$$
• 80% Confidence Level: $$C=80\%\to \frac{\alpha }{2}=0.10\to z_{.10}=|invNorm(0.10)|=1.282$$

Observation

As the confidence level increases, the margin of error also increases. This means we gain confidence but lose precision.

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Why Confidence Intervals?

Confidence intervals provide a range within which we expect the true population parameter to lie with a certain level of confidence. This is more informative than a single point estimate, as it accounts for sampling variability.

For example:

• If $\hat{p}=0.45$ with $E=0.03$, the 95% confidence interval is: $$0.45\pm 0.03=[0.42,0.48]$$ This means we are 95% confident the true population proportion lies within this range.