Inference for a Single Proportion

Confidence Intervals and Hypothesis Tests

Warm-Up Problem

Use the Central Limit Theorem to answer the following question.

Scenario (Polling Results): In a past election, 55% of voters supported Candidate A. A polling organization takes a random sample of 500 voters and finds that 53% of them support Candidate A.

If the true proportion of support has remained at 55%, what is the probability that a random sample of 500 voters would show 53% or less support for Candidate A?

In Excel: =NORM.DIST(0.53, 0.55, SQRT(0.55*0.45/500), TRUE)

The Highlights

• A brief reminder on proportions and the success-failure condition
• Confidence Intervals for a Population Proportion
• The general formula
• A worked example
• Hypothesis Tests for a Population Proportion
• The 8-step framework
• A worked example
• Practice problems

A Reminder on Proportions

When Can We Use the Normal Approximation?

Confidence Intervals: The General Formula

A First Confidence Interval

Ask Me At Least Two Questions

• Constructing a confidence interval involves several steps — identifying the point estimate, checking conditions, computing the standard error, finding the critical value, and interpreting the result
• This is a procedure that takes time and practice to feel natural
• Very few people feel comfortable with statistical inference quickly — ask questions early and often

Intuition for Testing Claims

The Hypothesis Testing Framework

The Hypothesis Testing Framework (continued)

Hypothesis Testing: A Worked Example

Hypothesis Testing: A Worked Example

Hypothesis Testing: A Worked Example

Hypothesis Testing: A Worked Example

Hypothesis Testing: A Worked Example

Conclusion in Context: The observed sample data do not provide significant evidence to suggest that the proportion of patients experiencing the adverse side effect is lower than 10%.

Ask Me At Least Two Questions

• The hypothesis testing procedure has many steps — and the details matter at each one
• This is a procedure that will take time and practice to master
• Some common stumbling points:
• Confusing the direction of the alternative hypothesis (left, right, or two-tailed)
• Using p̂ instead of p₀ in the standard error for the test statistic
• Forgetting to double the tail probability for a two-tailed test
• Conflating "fail to reject H₀" with "accept H₀" — we never prove H₀ is true
• Ask questions early and often

Practice: Influencer and Viral Videos

Scenario: A social media influencer claims that at least 60% of their followers have watched their latest viral video. A survey of 400 randomly selected followers finds that 54% have watched it. Is there enough evidence to reject the influencer's claim at the 5% significance level?

Practice: Fast Food Preferences

Scenario: A random sample of 300 fast-food enthusiasts shows that 65% prefer a chain's chicken sandwich over competitors'. Construct a 90% confidence interval for the true proportion who prefer the sandwich. Interpret the interval in context.

Practice: Vaccine Approval Rate

Scenario: A vaccine was found effective in 85% of clinical trial cases. A post-approval survey of 1,200 recipients finds that 83% report effectiveness. Is this evidence that the true effectiveness rate differs from 85%? Test at the 1% significance level.

Practice: Coffee Shop Loyalty Cards

Scenario: A coffee shop claims 40% of customers use loyalty cards. A sample of 250 customers reveals 38% do. At the 10% significance level, does the sample provide evidence that fewer than 40% use loyalty cards?

Practice: Recycling Habits in a City

Scenario: A local survey of 800 randomly selected households finds that 68% recycle. Construct and interpret a 90% confidence interval for the true proportion of recyclers. A city initiative aims to raise recycling rates to at least 65% — what does your interval suggest about the initiative's success?

Practice: Return Rates for an Online Store

Scenario: An online store audits 225 randomly selected transactions and finds a 6.5% return rate. Construct and interpret a 95% confidence interval for the true return rate.

Practice: Video Game Controller Popularity

Scenario: A limited release of a new controller design to 93 gamers finds that 79.6% prefer the new design. Construct and interpret a 98% confidence interval. The company will mass-produce if at least 75% prefer the new design — what does your interval suggest?

Practice: Support for a New Environmental Policy

Scenario: A politician claims that a minority of the public supports a new environmental policy. A sample of 627 people finds that 54% support it. At the 10% significance level, can you reject the politician's claim?

Exit Ticket

Navigate to our MAT240 Exit Ticket Form, answer the questions, and complete the task below.

Note. Today’s discussion is listed as 13. Inference on a Proportion

Task: A researcher wants to know whether a new fertilizer increases average crop yield, and by how much, compared to the standard fertilizer. Would a confidence interval or a hypothesis test better answer their question? Justify your answer, and describe what information the other approach would fail to provide.

Next Time…

• What we'll be doing…
• Mixed practice with inference for proportions — both confidence intervals and hypothesis tests
• Inference involving single proportions and comparisons between proportions
• Homework: Complete the Practice Inference for Proportions interactive prep-work activity and submit the hash codes using the Google Form from this week’s BrightSpace announcement.