One-sample confidence interval

Definition

A one-sample confidence interval is a statistical range that estimates the true mean of a population based on a sample drawn from that population. This interval provides a margin of error around the sample mean, giving an idea of how much uncertainty is involved in estimating the population parameter. The width of the interval depends on factors like the sample size, variability in the data, and the desired confidence level.

5 Must Know Facts For Your Next Test

1. A one-sample confidence interval is typically calculated using the formula: sample mean ± (critical value) × (standard error).
2. The critical value is determined based on the desired confidence level and can be found using statistical tables like z-tables or t-tables.
3. The standard error represents how much variability can be expected in sample means and is calculated as the standard deviation divided by the square root of the sample size.
4. As sample size increases, the width of the one-sample confidence interval tends to decrease, leading to a more precise estimate of the population mean.
5. Common confidence levels used are 90%, 95%, and 99%, with higher confidence levels resulting in wider intervals to ensure greater certainty.

Review Questions

• How do you interpret a one-sample confidence interval if you have a sample mean of 50 with a 95% confidence interval of (45, 55)?
  • A one-sample confidence interval of (45, 55) with a sample mean of 50 means we are 95% confident that the true population mean lies between 45 and 55. This implies that if we were to take many samples and calculate their confidence intervals, approximately 95% of those intervals would contain the actual population mean. It highlights both the estimate and the uncertainty around it.
• What factors influence the width of a one-sample confidence interval, and how do they interact with each other?
  • The width of a one-sample confidence interval is influenced by three main factors: sample size, variability in data, and confidence level. A larger sample size reduces variability in estimates, leading to a narrower interval. Conversely, higher variability increases uncertainty, resulting in a wider interval. Lastly, choosing a higher confidence level also broadens the interval to ensure more certainty about capturing the true population mean.
• Evaluate how changing from a 95% to a 99% confidence level affects a one-sample confidence interval and discuss its practical implications.
  • Changing from a 95% to a 99% confidence level will widen the one-sample confidence interval because it requires greater assurance that the true population mean is included. Practically, this means while we gain more certainty about our estimate, we also lose precision since our estimates become less specific. This trade-off is crucial when deciding how confident we want to be about our estimates versus how precise we need them to be for decision-making.

Related terms

Sample Mean: The average value calculated from a subset of data points taken from a larger population, used as a point estimate for the population mean.

Confidence Level: The probability that the confidence interval contains the true population parameter, commonly expressed as a percentage such as 95% or 99%.

Margin of Error: The range of values above and below the sample mean that defines the width of the confidence interval, reflecting the level of uncertainty associated with the sample estimate.