6.2 Confidence Interval for the Mean (σ unknown)

Front

What is a confidence interval?

Back

A confidence interval is a range of values, derived from a data set, that is likely to contain the value of an unknown population parameter, such as the mean, with a specified level of confidence.

Front

What does it mean when we say a confidence interval is 95%?

Back

A 95% confidence interval means that if we were to take many samples and build a confidence interval from each sample, approximately 95% of those intervals would contain the true population mean.

Front

What is the formula for calculating a confidence interval for the mean when the population standard deviation is unknown?

Back

The formula is: CI = x̄ ± t*(s/√n), where x̄ is the sample mean, t is the t-score from the t-distribution, s is the sample standard deviation, and n is the sample size.

Front

What is the t-distribution?

Back

The t-distribution is a probability distribution that is used to estimate population parameters when the sample size is small and/or the population standard deviation is unknown.

Front

How do you determine the t-score for a confidence interval?

Back

The t-score can be determined using a t-table, which requires the desired confidence level and the degrees of freedom (df = n - 1, where n is the sample size).

Front

What is the significance of degrees of freedom in confidence intervals?

Back

Degrees of freedom are used in statistical calculations to account for the number of independent values in a sample. In confidence intervals, it affects the t-score used.

Front

What is the relationship between sample size and confidence interval width?

Back

As the sample size increases, the width of the confidence interval decreases, leading to a more precise estimate of the population mean.

Front

What is the effect of increasing the confidence level on the confidence interval?

Back

Increasing the confidence level (e.g., from 95% to 99%) results in a wider confidence interval, as we require a larger range to be more certain that it contains the population mean.

Front

How do you interpret a confidence interval?

Back

A confidence interval provides a range of plausible values for the population mean. For example, if the interval is (2803, 3437), we are confident that the true mean lies within this range.

Front

What is the standard error of the mean (SEM)?

Back

The standard error of the mean is the standard deviation of the sampling distribution of the sample mean, calculated as SEM = s/√n.