AP Statistics: Confidence Intervals for Proportions - The Night Before 🌃

Hey! Let's get you totally prepped for tomorrow's AP Stats exam. We're focusing on confidence intervals for proportions – a key topic that often pops up. This guide is designed to be super clear, quick to navigate, and packed with everything you need to feel confident. Let's do this! 💪

What's a Confidence Interval?

A confidence interval is a range of values, calculated from sample data, used to estimate a population parameter (like a population proportion). Think of it as an educated guess with a built-in margin for error. For categorical data, we're estimating the true proportion of a population that has a specific characteristic. 🎯

• It's built from:

  • The sample proportion (our best guess from the data).
  • The sample size (how much data we have).
  • The sampling distribution (how sample proportions behave).

• Confidence Level: This tells us how sure we are that our interval contains the true population proportion. A 95% confidence level is super common. 📈

Checking Conditions: The Gatekeepers 🚪

Before we even think about calculating a confidence interval, we must check these conditions. No exceptions! These ensure our interval is valid and reliable.

1. Random Sample

• Why? Reduces bias. If our sample isn't random, our results can't be generalized to the larger population. 🚫

- **Scope of Inference:** Without randomness, our scope of inference is limited. We can't say our findings apply to the whole population.

2. Independence

• Why? We need to ensure that one subject's data doesn't influence another's. 👯

• The 10% Rule: When sampling without replacement, check if our sample size (n) is less than or equal to 10% of the population size (N). n ≤ 10%N

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3. Normal Condition

• Why? Allows us to use the normal distribution (z-scores) for calculations. 📊
• Large Counts Condition: We need at least 10 expected successes and 10 expected failures.

  • Check: np ≥ 10 AND n(1-p) ≥ 10

  • Where:

    • n = sample size
    • p = estimated sample proportion

-   🎉 If all conditions are met, we can use the normal distribution! 🎉

One-Sample z-Interval for Proportions

This is the specific procedure we use to build a confidence interval for a single population proportion. It relies on the z-distribution. 🤓

• When to Use It:
  • We have a random sample (or large enough sample for normal approximation).
  • We're dealing with a single categorical variable (yes/no, etc.).
  • The population proportion is unknown and we need to estimate it.

Calculating the Interval: Point Estimate + Margin of Error

Every confidence interval has two main parts:

1. Point Estimate

• This is our best single guess for the population proportion.
• It's the sample proportion (p̂). 📱
• It's the center of our confidence interval.

2. Margin of Error

• This is our "buffer zone" to account for uncertainty. Think of it as how much we add and subtract from our point estimate to get the interval's bounds.

• It's based on:

  • The critical value (z-score): Determined by our confidence level (e.g., 1.96 for 95%).
  • The standard error of the proportion: Variability of the sample proportion.

• High-Value Topic: Sample size matters! Larger sample sizes = smaller standard error = smaller margin of error = more precise estimate.

Formula:

$$\text{Confidence Interval} = \hat{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$

Where: - $\hat{p}$ is the sample proportion - $z^*$ is the critical value (z-score) - n is the sample size

Rearranging for Sample Size

• We can rearrange the margin of error formula to find the minimum sample size needed for a specific margin of error:

  $$n = \left( \frac{z^*}{ME} \right)^2 \cdot p(1-p)$$

  • Where:

    • n = minimum sample size
    • z = critical value
    • ME = desired margin of error
    • p = estimated proportion (use 0.5 for maximum sample size)

❤️

Using a Calculator

• TI-84 Shortcut: Go to STAT -> TESTS -> 1-PropZInt.
• Enter:
  • x = number of successes
  • n = sample size
  • C-level = confidence level
• Hit Calculate and boom! You have your interval. 🖥️

Final Exam Focus

Alright, let's focus on what's most important for the exam. These are the areas where you'll likely see the most questions:

• Conditions: Randomness, Independence (10% rule), Normality (Large Counts). Nail these down! 💯
• Interpretation: Understand what the confidence level actually means. It's the success rate of the method, not the probability of the interval containing the true parameter.
• Impact of Sample Size & Confidence Level: Know how these affect the width of the interval. ⬆️
• Calculator Skills: Be fast and accurate with your calculator for z-intervals. ⏱️

Last-Minute Tips:

• Time Management: Don't spend too long on any one question. Move on and come back if you have time.
• Common Pitfalls:
  • Forgetting to check conditions.
  • Confusing confidence level with the probability of the interval containing the true parameter.
  • Misinterpreting the meaning of a confidence interval.
• FRQ Strategies:
  • Show all your work. Even if you make a mistake, you can get partial credit.
  • Write in context. Use the language of the problem in your explanations.
  • Answer the question directly. Don't go on tangents.

Practice Questions

Let's test your knowledge with some practice questions that mimic what you might see on the AP exam.

Practice Question

Multiple Choice Questions

1. A polling organization surveys 500 randomly selected registered voters and finds that 280 of them plan to vote for Candidate A. Which of the following is the most appropriate 95% confidence interval for the proportion of all registered voters who plan to vote for Candidate A?

   (A) 0.56 ± 1.96 * sqrt((0.56 * 0.44) / 500) (B) 0.56 ± 1.645 * sqrt((0.56 * 0.44) / 500) (C) 0.56 ± 1.96 * sqrt((0.56) / 500) (D) 0.56 ± 1.96 * sqrt((0.28 * 0.72) / 500) (E) 0.56 ± 1.96 * sqrt((0.28 * 0.72) / 280)

2. A researcher wants to estimate the proportion of adults who support a new policy. They want to create a 99% confidence interval with a margin of error of no more than 0.04. Assuming they have no prior estimate of the proportion, what is the minimum sample size they should use?

   (A) 415 (B) 830 (C) 1037 (D) 1657 (E) 2074

Free Response Question

A local high school is considering implementing a new policy regarding cell phone use during class. They conduct a survey of a random sample of 200 students and find that 120 of them support the new policy.

(a) Construct and interpret a 90% confidence interval for the proportion of all students at the school who support the new policy.

(b) Based on the interval you constructed in part (a), is it plausible that a majority of students at the school support the new policy? Explain.

(c) Suppose the school administration wants to conduct another survey to estimate the proportion of students who support the new policy with a margin of error of no more than 0.03 with 95% confidence. How many students should they survey? Assume they have no prior estimate of the proportion.

Scoring Guide

(a) Construct and interpret a 90% confidence interval for the proportion of all students at the school who support the new policy.

• Conditions (1 point):
  • Random: Stated as random sample
  • Independence: It is reasonable to assume there are more than 2000 students at the school
  • Normality: np = 200(0.6) = 120 ≥ 10 and n(1-p) = 200(0.4) = 80 ≥ 10
• Calculations (2 points):
  • Correct interval formula: 0.6 ± 1.645 * sqrt((0.6 * 0.4) / 200)
  • Interval: (0.543, 0.657)
• Interpretation (1 point):
  • We are 90% confident that the true proportion of students who support the policy is between 0.543 and 0.657

(b) Based on the interval you constructed in part (a), is it plausible that a majority of students at the school support the new policy? Explain.

• Answer (1 point):

  • Yes, it is plausible because the entire interval is above 0.5. (c) Suppose the school administration wants to conduct another survey to estimate the proportion of students who support the new policy with a margin of error of no more than 0.03 with 95% confidence. How many students should they survey? Assume they have no prior estimate of the proportion.

• Calculations (2 points):

  • Correct formula: n = (1.96/0.03)^2 * 0.5 * 0.5
  • Sample size: 1067.11. Since we need a whole number, we round up to 1068.

You've got this! Go get that 5! 🚀