Glossary
10% Rule
A condition for independence when sampling without replacement, stating that the sample size (n) must be no more than 10% of the population size (N) to ensure that the probability of selecting an item doesn't change significantly.
Example:
If we sample 50 students from a high school, the 10% Rule requires that the school must have at least 500 students for our sample to be considered independent.
Confidence Interval
A range of values, calculated from sample data, used to estimate an unknown population parameter, such as a population proportion.
Example:
After surveying students, we constructed a confidence interval of (0.52, 0.68) for the proportion of students who prefer online learning, suggesting the true proportion is likely within this range.
Confidence Level
The percentage that expresses how sure we are that our confidence interval contains the true population parameter. A higher confidence level results in a wider interval.
Example:
A 95% confidence level means that if we were to repeat this sampling process many times, about 95% of the resulting intervals would capture the true population proportion.
Critical Value (z-score)
A value from the standard normal distribution that corresponds to a specific confidence level, used to calculate the margin of error.
Example:
For a 95% confidence interval, the critical value (z*) is 1.96, which marks the boundary for the middle 95% of the standard normal distribution.
Independence
A condition for inference stating that the outcome of one observation does not influence the outcome of another, ensuring that observations provide unique information.
Example:
When flipping a coin multiple times, each flip is independent of the others; the result of one flip doesn't affect the next.
Margin of Error
The range of values above and below the point estimate in a confidence interval, accounting for the uncertainty due to sampling variability.
Example:
A survey reports a 55% approval rating with a margin of error of ±3%, meaning the true approval rating is likely between 52% and 58%.
Normal Condition (Large Counts Condition)
A condition for using the normal distribution to model the sampling distribution of a sample proportion, requiring at least 10 expected successes and 10 expected failures.
Example:
Before constructing a confidence interval for the proportion of students who own a laptop, we must check the Normal Condition by ensuring that both np̂ and n(1-p̂) are at least 10.
One-Sample z-Interval for Proportions
The specific statistical procedure used to construct a confidence interval for a single unknown population proportion, relying on the standard normal (z) distribution.
Example:
To estimate the true proportion of defective items in a large shipment, a quality control engineer would use a One-Sample z-Interval for Proportions.
Point Estimate
A single value, calculated from sample data, that serves as the best single guess for an unknown population parameter.
Example:
If 60 out of 100 surveyed students prefer chocolate ice cream, then 0.60 is our point estimate for the true proportion of all students who prefer chocolate ice cream.
Random Sample
A sample selected in such a way that every individual or item in the population has an equal chance of being chosen, which helps reduce bias and allows for generalization to the population.
Example:
To estimate the average height of students, a researcher used a computer to randomly select 100 students from the school's enrollment list, ensuring a random sample.
Sample Size (for desired ME)
The minimum number of observations needed in a sample to achieve a specified margin of error at a given confidence level, often calculated using a rearranged margin of error formula.
Example:
To ensure a margin of error of no more than 0.02 for a 99% confidence interval, we need to calculate the required sample size before conducting the survey.
Standard Error of the Proportion
An estimate of the standard deviation of the sampling distribution of the sample proportion, indicating the typical variability of sample proportions around the true population proportion.
Example:
A smaller standard error of the proportion suggests that our sample proportion is likely a more precise estimate of the true population proportion.