95% confidence interval

Definition

A 95% confidence interval is a statistical range that estimates where a population parameter, such as a mean or proportion, is likely to fall with a 95% level of certainty. This means that if we were to take many samples and calculate a confidence interval from each sample, about 95% of those intervals would contain the true population parameter. This concept is crucial in making inferences based on sample data and is often used in conjunction with hypothesis testing to assess the reliability of results.

5 Must Know Facts For Your Next Test

1. A 95% confidence interval is typically calculated using the formula: $$ ext{Point Estimate} \pm (Z^* \times \text{Standard Error})$$ where $$Z^*$$ is the critical value for a normal distribution corresponding to 95% confidence.
2. When constructing confidence intervals for means, if the sample size is large (usually n > 30), the z-distribution can be used; for smaller samples, the t-distribution is more appropriate.
3. The width of a confidence interval is influenced by the sample size and variability in the data; larger samples tend to produce narrower intervals.
4. A wider confidence interval indicates less precision in estimating the population parameter, while a narrower interval suggests more precise estimates.
5. The 95% confidence level implies that there is a 5% chance that the true population parameter does not lie within the interval, which reflects the trade-off between confidence and precision.

Review Questions

• How does increasing the sample size affect the width of a 95% confidence interval?
  • Increasing the sample size decreases the width of a 95% confidence interval. This occurs because a larger sample provides more information about the population, resulting in a smaller standard error. As the formula for calculating the confidence interval includes standard error, reducing this value leads to a narrower interval, indicating a more precise estimate of the population parameter.
• Discuss how a 95% confidence interval can be used to interpret results from hypothesis testing.
  • A 95% confidence interval provides a range of plausible values for a population parameter and can help inform hypothesis testing outcomes. If the null hypothesis value falls outside this interval, it suggests that we can reject the null hypothesis at the 0.05 significance level, indicating that our sample data provides sufficient evidence against it. Conversely, if the null hypothesis value lies within the interval, we fail to reject it, implying that our sample does not provide strong enough evidence to support an alternative claim.
• Evaluate how different significance levels influence the construction of confidence intervals and their interpretation.
  • Different significance levels, such as 90%, 95%, or 99%, lead to varying widths in confidence intervals. A lower significance level results in a narrower interval but less certainty about containing the true parameter, while a higher significance level creates a wider interval with greater certainty. This balance influences decision-making in hypothesis testing; choosing a significance level affects not only how conservative or liberal our tests are but also how we interpret the reliability of our estimates regarding population parameters.

Related terms

Margin of Error: The amount of error allowed in the estimation of a population parameter, calculated as part of the confidence interval.

Point Estimate: A single value derived from sample data that serves as the best estimate of a population parameter.

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