Activity 31

MATH 216: Statistical Thinking

Proportion Inference Practice

Time Allocation: 15 minutes total

Scenario 1: Political Polling

A political poll tests if candidate support differs from the historical baseline of 50%. Survey of 400 voters found 192 supporters.

Sample size: n = 400 
Number of supporters: x = 192 
Sample proportion: p̂ = 0.48

R proportion test output:

1-sample proportions test without continuity correction

data:  x out of n, null probability p_0
X-squared = 0.64, df = 1, p-value = 0.4237
alternative hypothesis: true p is not equal to 0.5
95 percent confidence interval:
 0.433 0.527
sample estimates:
   p
0.48

Questions: - State the null and alternative hypotheses - What is the statistical decision at α = 0.05? - Interpret the confidence interval

Scenario 2: Quality Control

A manufacturer tests if their new process reduces defect rate below the historical 10%. In a batch of 500 products, 40 were defective.

Sample size: n = 500 
Defective products: x = 40 
Defect rate: p̂ = 0.08

R proportion test output:

1-sample proportions test without continuity correction

data:  x out of n, null probability p_0
X-squared = 2.2222, df = 1, p-value = 0.068
alternative hypothesis: true p is less than 0.1
95 percent confidence interval:
 0.0000000 0.096
sample estimates:
   p
0.08

Questions: - State the null and alternative hypotheses - What is the statistical decision at α = 0.05? - What practical recommendation would you make?

Scenario 3: Customer Satisfaction Survey

A company tests if customer satisfaction exceeds 80%. Survey of 250 customers found 210 satisfied.

Sample size: n = 250 
Satisfied customers: x = 210 
Satisfaction rate: p̂ = 0.84

R proportion test output:

1-sample proportions test without continuity correction

data:  x out of n, null probability p_0
X-squared = 2.5, df = 1, p-value = 0.057
alternative hypothesis: true p is greater than 0.8
95 percent confidence interval:
 0.795 1.000
sample estimates:
   p
0.84

Questions:

  • State the null and alternative hypotheses
  • What is the statistical decision at α = 0.05?
  • Interpret the confidence interval

Scenario 4: Medical Treatment Efficacy

A clinical trial tests if a new treatment has a success rate different from the standard treatment’s 65% success rate. In a study of 300 patients, 195 showed improvement.

Sample size: n = 300 
Patients improved: x = 195 
Success rate: p̂ = 0.65

R proportion test output:

1-sample proportions test without continuity correction

data:  x out of n, null probability p_0
X-squared = 0, df = 1, p-value = 1
alternative hypothesis: true p is not equal to 0.65
95 percent confidence interval:
 0.596 0.704
sample estimates:
   p
0.65

Questions:

  • State the null and alternative hypotheses
  • What is the statistical decision at α = 0.05?
  • Interpret the confidence interval
  • What does the p-value of 1 indicate about the sample proportion?