What is the confidence interval for the mean and standard deviation?

A confidence interval provides a range of values within which the true population mean or standard deviation is likely to lie.

Confidence Interval for the Mean

Definition: The confidence interval for the mean is a range of values that is likely to contain the true population mean.
Formula: The formula for calculating the confidence interval for the mean depends on whether the population standard deviation is known or unknown.
- Known population standard deviation:
  - CI = x̄ ± z(σ/√n)
- Unknown population standard deviation:
  - CI = x̄ ± t(s/√n)
Interpretation: The confidence interval for the mean is interpreted as follows: We are [confidence level]% confident that the true population mean lies within this interval.

Definition: The confidence interval for the standard deviation is a range of values that is likely to contain the true population standard deviation.
Formula: The formula for calculating the confidence interval for the standard deviation is more complex and involves the chi-square distribution.
Interpretation: The confidence interval for the standard deviation is interpreted as follows: We are [confidence level]% confident that the true population standard deviation lies within this interval.

Example 1: A researcher wants to estimate the average height of all students in a university. They take a sample of 100 students and find that the average height is 170 cm with a standard deviation of 10 cm. They want to calculate a 95% confidence interval for the mean height. Using the formula for known population standard deviation, they calculate the confidence interval to be (168.04 cm, 171.96 cm). This means that they are 95% confident that the true average height of all students in the university lies between 168.04 cm and 171.96 cm.
Example 2: A company wants to estimate the variability of the weight of its products. They take a sample of 20 products and find that the sample standard deviation is 2 grams. They want to calculate a 90% confidence interval for the standard deviation. Using the formula for the chi-square distribution, they calculate the confidence interval to be (1.5 grams, 2.8 grams). This means that they are 90% confident that the true standard deviation of the weight of all products lies between 1.5 grams and 2.8 grams.

Confidence intervals are widely used in statistical inference to estimate population parameters.
The width of the confidence interval depends on the confidence level, sample size, and variability of the data.
A larger sample size will result in a narrower confidence interval, providing a more precise estimate of the population parameter.
A higher confidence level will result in a wider confidence interval, indicating greater certainty in the estimate.