Confidence Intervals

Definition

A confidence interval is a range of values that is likely to contain an unknown population parameter with a specified degree of confidence. It provides a measure of the precision of an estimate and is used to make inferences about the true value of a parameter based on a sample statistic.

5 Must Know Facts For Your Next Test

1. Confidence intervals are used to quantify the uncertainty associated with a point estimate and provide a range of plausible values for the true population parameter.
2. The level of confidence, typically 90%, 95%, or 99%, represents the probability that the true parameter value will fall within the calculated interval.
3. Wider confidence intervals indicate less precision in the estimate, while narrower intervals suggest more reliable estimates.
4. The margin of error in a confidence interval is determined by the sample size, the variability in the data, and the desired level of confidence.
5. Confidence intervals are essential for making inferences and drawing conclusions about a population based on sample data.

Review Questions

• Explain how confidence intervals are used in the context of the Normal Distribution—Lap Times topic.
  • In the Normal Distribution—Lap Times topic, confidence intervals can be used to estimate the true mean lap time of a population based on a sample of lap times. The sample mean lap time would be the point estimate, and the confidence interval would provide a range of values that is likely to contain the true population mean lap time with a specified level of confidence, such as 95%. This allows researchers to quantify the uncertainty in the estimate and make inferences about the typical lap times in the population.
• Describe how the Central Limit Theorem (Pocket Change) topic is related to the use of confidence intervals.
  • The Central Limit Theorem (Pocket Change) topic is relevant to the use of confidence intervals because it states that the sampling distribution of the sample mean will be normally distributed, even if the underlying population distribution is not normal. This property allows researchers to construct reliable confidence intervals for the population mean, regardless of the shape of the original population distribution. By leveraging the Central Limit Theorem, researchers can make valid inferences about the true population mean of pocket change based on sample data and confidence intervals.
• Evaluate how confidence intervals can be applied in the Regression (Distance from School) topic to draw conclusions about the relationship between variables.
  • In the Regression (Distance from School) topic, confidence intervals can be used to quantify the uncertainty in the estimated regression coefficients, which represent the relationships between the predictor variables (e.g., distance from school) and the response variable (e.g., student achievement). By constructing confidence intervals for the regression coefficients, researchers can assess the statistical significance of the relationships and make inferences about the true population parameters. This allows them to draw conclusions about the strength and direction of the associations between the variables, while accounting for the inherent variability in the sample data.

Related terms

Point Estimate: A single value that represents the best estimate of a population parameter based on sample data.

Margin of Error: The range of values above and below the point estimate that defines the confidence interval.

Significance Level: The probability of rejecting the null hypothesis when it is true, commonly denoted as α.