AP Statistics 6.3 Justifying a Claim Based on a Confidence Interval for a Population Proportion Study Notes
AP Statistics 6.3 Justifying a Claim Based on a Confidence Interval for a Population Proportion Study Notes- New syllabus
AP Statistics 6.3 Justifying a Claim Based on a Confidence Interval for a Population Proportion Study Notes -As per latest AP Statistics Syllabus.
LEARNING OBJECTIVE
- An interval of values should be used to estimate parameters, in order to account for uncertainty
Key Concepts:
- Interpreting a Confidence Interval for a Population Proportion
- Justifying a Claim Using a Confidence Interval
- Relationships Between Sample Size, Confidence Level, Margin of Error, and Interval Width
Interpreting a Confidence Interval for a Population Proportion
Interpreting a Confidence Interval for a Population Proportion
What a Confidence Interval Means:
- A confidence interval provides a range of plausible values for the true population proportion, based on sample data.
- For example, a 95% confidence interval means: “We are 95% confident that the interval we calculated from our sample data contains the true population proportion.”
- Each individual interval either contains the population proportion or it does not — the randomness comes from the sampling process, not from the parameter itself.
Confidence Level (C%):
- The confidence level tells us how often the method succeeds in the long run.
- If we repeated the same sampling process many times and built a confidence interval each time, approximately C% of those intervals would contain the true population proportion.
Requirements for a Good Interpretation:
- Always connect the interval to the population proportion, not the sample.
- State the interval in terms of the context of the problem (e.g., students, voters, customers).
- Do not use probability language for the parameter (avoid saying “There is a 95% chance the true proportion is in this interval”).
Common Mistakes to Avoid:
- Wrong : “There is a 95% probability that the population proportion is between 0.42 and 0.50.”
Correct: The probability applies to the method, not the parameter. The true proportion is fixed. - Wrong : “95% of the sample proportions are between 0.42 and 0.50.”
Correct: The interval refers to the population parameter, not the sample data. - Wrong : “95% of students own between 42% and 50% of cars.”
Correct: Interpret in terms of the proportion of students, not a confusing restatement.
Example:
A 95% confidence interval for the proportion of students at a university who own a car is (0.42, 0.50).
▶️ Answer / Explanation
Correct Interpretation: We are 95% confident that between 42% and 50% of all students at the university own a car.
Why this is correct: The statement references the population (all students at the university), uses the correct confidence interpretation, and avoids probability language about the fixed parameter.
Example:
A poll of 600 adults found that 333 supported a new recycling program, giving a 90% confidence interval of (0.49, 0.55).
▶️ Answer / Explanation
Correct Interpretation: We are 90% confident that between 49% and 55% of all adults in the population support the new recycling program.
Note: This does not mean there is a 90% probability the true proportion is in the interval. Instead, if we took many random samples and constructed a confidence interval each time, about 90% of those intervals would contain the true population proportion.
Justifying a Claim Using a Confidence Interval
Justifying a Claim Using a Confidence Interval
A confidence interval for a population proportion provides a range of plausible values for the true proportion. We can use this interval to assess whether a specific claim about the population proportion is consistent with the data.
Steps to Justify a Claim:
- State the claim: Identify the population proportion being claimed.
- Compare with interval: Check whether the claimed value falls inside or outside the confidence interval.
- Draw a conclusion:
- If the claim is inside the interval → the sample data are consistent with the claim.
- If the claim is outside the interval → the sample data do not support the claim (unlikely to be true at the given confidence level).
Important: Just because a claim is inside the interval does not prove it is true — it only means the data are consistent with the claim.
Example:
A school newspaper claims that 60% of students support building a new gym. A survey of 300 students produces a 95% confidence interval of (0.53, 0.61) for the true proportion who support the project. Does the data support the claim?
▶️ Answer / Explanation
Step 1: The claim is \( p = 0.60 \).
Step 2: The interval is (0.53, 0.61). The claimed value 0.60 is inside the interval.
Step 3: Since 0.60 is a plausible value, the claim is supported by the data. (We cannot prove it, but the evidence is consistent with the claim.)
Example:
A company claims that at least 75% of customers are satisfied with their service. A random sample of 400 customers gives a 95% confidence interval of (0.68, 0.73). Is the company’s claim supported?
▶️ Answer / Explanation
Step 1: The claim is \( p \geq 0.75 \).
Step 2: The interval is (0.68, 0.73). The entire interval is below 0.75.
Step 3: Since 0.75 is not inside the interval, the claim is not supported by the data.
Relationships Between Sample Size, Confidence Level, Margin of Error, and Interval Width
Relationships Between Sample Size, Confidence Level, Margin of Error, and Interval Width
1. Effect of Sample Size (n):
- The standard error is \( SE = \sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}} \).
- As sample size increases, the denominator \( n \) increases → SE decreases → margin of error decreases → the confidence interval becomes narrower.
- Larger samples give more precise estimates.
2. Effect of Confidence Level (C%):
- The margin of error is \( ME = z^* \cdot SE \).
- As confidence level increases, \( z^* \) increases (for 95%, \( z^* = 1.96 \); for 99%, \( z^* = 2.576 \)).
- This makes the interval wider, since we require more confidence that the true proportion is captured.
3. Relationship Between Width and Margin of Error:
- The total width of the confidence interval is exactly twice the margin of error.
- Width = \( (\hat{p} + ME) – (\hat{p} – ME) = 2 \times ME \).
Summary:
- ↑ Sample size → ↓ Width of CI
- ↑ Confidence level → ↑ Width of CI
- Width of CI = 2 × Margin of Error
Example:
A survey finds that 60% of respondents favor a policy, based on a sample of 400 people. The 95% confidence interval is (0.556, 0.644).
Suppose the sample size was increased to 1600, while keeping everything else the same. What would happen to the width of the confidence interval?
▶️ Answer / Explanation
Step 1: Current SE = \( \sqrt{\dfrac{0.6(0.4)}{400}} = 0.0245 \). ME = \( 1.96(0.0245) = 0.048 \). Width = 2 × 0.048 = 0.096.
Step 2: With \( n = 1600 \), SE = \( \sqrt{\dfrac{0.6(0.4)}{1600}} = 0.0122 \). ME = \( 1.96(0.0122) = 0.024 \). Width = 2 × 0.024 = 0.048.
Conclusion: Increasing the sample size from 400 to 1600 cut the width of the confidence interval in half, making the estimate more precise.