AP Statistics:: Inference Proportions

A CONFIDENCE INTERVAL for a population proportion has the general form:

- **A)** \(\hat{p} \times (\text{critical value})\)
- **B)** \(\hat{p} \pm (\text{critical value}) \times (\text{standard error})\)
- **C)** \(p \pm \hat{p}\)
- **D)** \(n \pm \hat{p}\)

The STANDARD ERROR used in a one-sample confidence interval for a proportion is:

- **A)** \(\sqrt{\dfrac{p(1-p)}{n}}\)
- **B)** \(\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}\)
- **C)** \(n\hat{p}\)
- **D)** \(\dfrac{\hat{p}}{n}\)

In a confidence interval, the CONFIDENCE LEVEL (e.g., 95%) refers to:

- **A)** The long-run PROPORTION of all possible random samples for which the resulting confidence interval procedure would capture the true population parameter
- **B)** The probability that this SPECIFIC interval contains the parameter
- **C)** The proportion of the sample that supports a particular conclusion
- **D)** The margin of error itself

A 95% confidence interval for a population proportion is calculated as \((0.42, 0.58)\). Which is the CORRECT interpretation?

- **A)** 95% of the sample data falls between 0.42 and 0.58
- **B)** 95% of all possible sample proportions equal exactly 0.50
- **C)** There is a 95% probability that \(\hat{p}\) falls in this interval
- **D)** We are 95% confident that the true population proportion lies between 0.42 and 0.58

Increasing the CONFIDENCE LEVEL (e.g., from 90% to 99%), while keeping the sample size the same, generally causes the confidence interval's WIDTH to:

- **A)** Become negative
- **B)** Decrease
- **C)** INCREASE, since a higher confidence level requires a larger critical value, widening the margin of error
- **D)** Stay exactly the same

Increasing the SAMPLE SIZE, while keeping the confidence level the same, generally causes the confidence interval's WIDTH to:

- **A)** Become undefined
- **B)** DECREASE, since a larger sample reduces the standard error, producing a more precise (narrower) interval
- **C)** Increase
- **D)** Stay exactly the same

The conditions required to construct a valid confidence interval for a population proportion include:

- **A)** The population must be exactly Normal
- **B)** Only that the sample size exceeds 1000
- **C)** RANDOM sample/assignment, the 10% condition (if sampling without replacement), and the Large Counts condition (\(n\hat{p} \ge 10\) and \(n(1-\hat{p}) \ge 10\))
- **D)** No conditions are required for proportion intervals

A researcher wants a NARROWER confidence interval without changing the confidence level. What is the most direct way to achieve this?

- **A)** Use a smaller critical value chosen arbitrarily
- **B)** Decrease the sample size
- **C)** Increase the confidence level
- **D)** Increase the sample size