PPT for Chapter 3 (Statistical Inference)

Confidence Intervals & Hypothesis Testing for Proportions

Slide 1: Title Slide

Source: https://wisc.pb.unizin.org/biocorestatistics/chapter/statistical-inference/

Slide 2: Core Definitions

Population
Complete set of interest

Sample
Observed subset for inference

Parameter
- \(p\) (proportion)
- \(\mu\) (mean)

Statistic
- \(\hat{p}\) (sample proportion)
- \(\bar{x}\) (sample mean)

Slide 3: Inference Process

graph LR
  A[Population] --> B[Random Sample]
  B --> C[Calculate Statistic]
  C --> D[Make Inference]

graph LR
  A[Population] --> B[Random Sample]
  B --> C[Calculate Statistic]
  C --> D[Make Inference]

Slide 4: Confidence Intervals for Proportions

\[ \hat{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \]

Slide 5: Political Poll Example

Scenario:

280 supporters out of 500 voters
95% confidence level

prop.test(280, 500, conf.level = 0.95, correct = FALSE)


    1-sample proportions test without continuity correction

data:  280 out of 500, null probability 0.5
X-squared = 7.2, df = 1, p-value = 0.00729
alternative hypothesis: true p is not equal to 0.5
95 percent confidence interval:
 0.5161969 0.6028882
sample estimates:
   p 
0.56

Slide 6: Hypothesis Testing

\[\text{Null Hypothesis} ~~~~H_0: p = p_0\] \[\text{Alternatives Hypothesis} ~~~~H_a: p < p_0~~~ (left.sided)\] \[\text{Alternatives Hypothesis} ~~~~H_a: p > p_0~~~ (right.sided)\] \[\text{Alternatives Hypothesis} ~~~~H_a: p \neq p_0~~~ (two.sided)\]

Slide 7: Quality Control Case

Scenario:

Claim: Defects > 5%
Found: 22 defects in 300

prop.test(22, 300, p = 0.05, 
                    alternative = "greater", 
                    correct = FALSE)


    1-sample proportions test without continuity correction

data:  22 out of 300, null probability 0.05
X-squared = 3.4386, df = 1, p-value = 0.03184
alternative hypothesis: true p is greater than 0.05
95 percent confidence interval:
 0.05220849 1.00000000
sample estimates:
         p 
0.07333333

Slide 8: Error Types