Unit Six: Inference for Categorical Data: Proportions- essential knowledge

What does it mean when data distributions are described as random or not?

Variation in the shapes of data distributions may be either random (due to chance) or non-random (due to systematic factors).

What is the appropriate confidence interval procedure for a one-sample proportion with one categorical variable?

The appropriate confidence interval procedure is a one-sample z-interval for a proportion.

What are the necessary assumptions for inference on population proportions, means, and slopes?

We must check for independence in data collection methods and select the appropriate sampling distribution.

What does the margin of error represent in statistical inference?

The margin of error represents how much the sample statistic is likely to vary from the corresponding population parameter.

How can the formula for margin of error be rearranged, and what is the purpose?

The formula can be rearranged to solve for the minimum sample size (n) needed to achieve a given margin of error.

How is a critical value related to confidence intervals in a standard normal distribution?

Critical values represent the boundaries that encompass the middle C% of the distribution, indicating the confidence level for a proportion.

How are confidence intervals used in estimating population proportions?

Confidence intervals provide interval estimates for the population proportion based on random sample data, which vary from sample to sample.

What does it mean to be C% confident in a confidence interval for a population proportion?

It means we are C% confident that the interval captures the true population proportion.

How should a confidence interval for a one-sample proportion be interpreted?

The interpretation should reference the sample taken and provide context about the population it represents.

What happens to the width of a confidence interval for a population proportion as the sample size increases?

The width of the confidence interval decreases as the sample size increases.

How does the confidence level affect the width of a confidence interval for a population proportion?

The width of the confidence interval increases as the confidence level increases.

What is the relationship between the margin of error and the width of a confidence interval for a population proportion?

The width of the confidence interval is exactly twice the margin of error.

What is the null hypothesis in hypothesis testing?

The null hypothesis is the assumption that there is no effect or difference unless evidence suggests otherwise.

How are null and alternative hypotheses typically structured?

The null hypothesis contains an equality reference (e.g., =, ≥, ≤), while the alternative hypothesis contains a strict inequality (e.g., <, >, ≠).

What is the null hypothesis for a population proportion?

The null hypothesis is H₀: p = p₀, where p₀ is the hypothesized population proportion.

What type of test is appropriate for a population proportion with a single categorical variable?

A one-sample z-test for a population proportion is appropriate.

How is a p-value interpreted in hypothesis testing for a population proportion?

The p-value represents the probability of obtaining a test statistic as extreme or more extreme than the observed value if the null hypothesis is true.

What is the significance level (α) in hypothesis testing?

The significance level is the predetermined probability of rejecting the null hypothesis when it is true (Type I error).

How do you decide whether to reject or fail to reject the null hypothesis?

Compare the p-value to the significance level (α). If p-value ≤ α, reject the null hypothesis; if p-value > α, fail to reject it.

What is a Type I error in hypothesis testing?

A Type I error occurs when the null hypothesis is true, but it is incorrectly rejected (false positive).

What is a Type II error in hypothesis testing?

A Type II error occurs when the null hypothesis is false, but it is not rejected (false negative).

How does increasing the sample size affect the probability of a Type II error?

Increasing the sample size reduces the probability of a Type II error.

What is the null hypothesis for the difference between two population proportions?

The null hypothesis is H₀: p₁ = p₂, or H₀: p₁ - p₂ = 0.

What test is appropriate for comparing two population proportions?

A two-sample z-test for the difference between two population proportions is appropriate.

How is the p-value interpreted in the context of testing the difference between two population proportions?

The p-value is interpreted assuming that the null hypothesis of equal proportions is true.