Confidence interval for the mean

Definition

A confidence interval for the mean is a range of values, derived from sample data, that is likely to contain the true population mean with a specified level of confidence. It provides a way to express the uncertainty about the estimate of the mean by incorporating both the sample mean and the variability in the data. The width of this interval reflects how precise the estimate is, which can be influenced by factors such as sample size and variability.

5 Must Know Facts For Your Next Test

1. The confidence interval is typically expressed in terms of a percentage, such as 95% or 99%, indicating the level of certainty that the true mean lies within that range.
2. A larger sample size will produce a narrower confidence interval because it reduces variability and provides a more accurate estimate of the population mean.
3. The formula for calculating a confidence interval for the mean when the population standard deviation is known is: $$ar{x} \\pm z \frac{\sigma}{\sqrt{n}}$$, where \( \bar{x} \) is the sample mean, \( z \) is the z-value corresponding to the desired confidence level, \( \sigma \) is the population standard deviation, and \( n \) is the sample size.
4. When the population standard deviation is unknown, the t-distribution is used instead of the normal distribution, resulting in slightly wider intervals.
5. Interpreting a confidence interval does not mean there is a specific probability that any one interval contains the true mean; rather, if we were to take many samples and create intervals from them, a certain percentage (e.g., 95%) would contain the true mean.

Review Questions

• How does increasing sample size affect the confidence interval for the mean and why?
  • Increasing sample size affects the confidence interval for the mean by making it narrower. A larger sample size reduces variability in the estimate of the mean due to better representation of the population, leading to a more accurate calculation. This results in less uncertainty about where the true population mean lies, thus providing more precision in our estimate.
• Discuss how the choice between using z-distribution or t-distribution influences confidence intervals for means.
  • The choice between using z-distribution and t-distribution significantly impacts confidence intervals for means based on whether we know the population standard deviation. If it is known and sample sizes are large (typically n > 30), we use z-distribution which gives us narrower intervals. However, if the population standard deviation is unknown or sample sizes are small, we should use t-distribution which accounts for additional uncertainty, resulting in wider intervals.
• Evaluate how changing from a 95% confidence level to a 99% confidence level affects the interpretation and application of confidence intervals for means.
  • Changing from a 95% to a 99% confidence level increases our certainty that the true population mean lies within our calculated interval. However, this comes at a cost: wider intervals. While a 99% confidence interval provides more assurance about capturing the true mean, it may also lead to less precise estimates and potentially less practical decision-making since it encompasses a broader range of values. This balance between precision and certainty needs to be carefully considered based on context.

Related terms

Sample Mean:

The average value calculated from a subset of data taken from a larger population, used as an estimate of the population mean.

Margin of Error:

The amount of error that is allowed in the results of a survey or experiment, which influences the width of the confidence interval.

Standard Error: An estimate of the standard deviation of the sampling distribution of a statistic, typically the mean, used to calculate confidence intervals.