What requirements must be satisfied in order to construct a confidence interval about a population mean?

Verified step by step guidance

1. Identify that constructing a confidence interval for a population mean typically requires certain assumptions about the data and the population distribution.
2. First, ensure that the sample is a simple random sample from the population, meaning each member of the population has an equal chance of being selected.
3. Second, check the population distribution: if the population is normally distributed, the confidence interval can be constructed regardless of sample size; if the population is not normal, the sample size should be sufficiently large (usually n \geq 30) to invoke the Central Limit Theorem.
4. Third, verify that the population standard deviation is either known or unknown. If unknown, use the sample standard deviation and the t-distribution; if known, use the z-distribution.
5. Finally, confirm that the data are independent and that the sample size is less than 10% of the population if sampling without replacement, to maintain independence.

Key Concepts

Random Sampling

Random sampling ensures that the data collected is representative of the population, reducing bias. This is essential for the validity of the confidence interval, as it relies on the assumption that the sample accurately reflects the population characteristics.

Normality or Large Sample Size

The sampling distribution of the sample mean should be approximately normal. This is guaranteed if the population is normal or if the sample size is large (usually n ≥ 30) by the Central Limit Theorem, allowing the use of normal or t-distributions to construct the interval.

Known or Unknown Population Standard Deviation

If the population standard deviation is known, the z-distribution is used to construct the confidence interval. If unknown, the sample standard deviation estimates it, and the t-distribution is applied, accounting for extra uncertainty in the estimate.