Confidence Interval

Definition

A confidence interval is a range of values that is likely to contain an unknown population parameter, such as a mean or proportion, with a specified level of confidence. It provides a way to quantify the uncertainty associated with estimating a population characteristic from a sample.

5 Must Know Facts For Your Next Test

1. Confidence intervals are used to estimate population parameters, such as the mean or proportion, based on sample data.
2. The level of confidence, typically 90%, 95%, or 99%, represents the probability that the true population parameter will be contained within the calculated interval.
3. The width of the confidence interval is determined by the standard error of the sample statistic and the chosen level of confidence.
4. Confidence intervals are used in hypothesis testing to determine if the null hypothesis should be rejected or not.
5. The Central Limit Theorem is a key concept in the construction of confidence intervals for sample means and proportions.

Review Questions

• Explain how the concept of a confidence interval is related to the topics of data, sampling, and variation in data and sampling.
  • The concept of a confidence interval is closely tied to the topics of data, sampling, and variation in data and sampling. Confidence intervals are used to quantify the uncertainty associated with estimating a population parameter from a sample. The variability in the sample data, as measured by the standard error, directly affects the width of the confidence interval. Additionally, the sampling method and the size of the sample influence the reliability and precision of the confidence interval. Understanding these relationships is crucial in interpreting the results of a confidence interval and drawing appropriate conclusions about the population.
• Describe how the Central Limit Theorem and the normal distribution are used in the construction of confidence intervals for sample means.
  • The Central Limit Theorem states that as the sample size increases, the sampling distribution of the sample mean approaches a normal distribution, regardless of the underlying distribution of the population. This property allows for the use of the normal distribution to construct confidence intervals for sample means. When the population standard deviation is known, the standard normal distribution can be used to calculate the margin of error and construct the confidence interval. When the population standard deviation is unknown, the Student's t-distribution is used instead. The Central Limit Theorem ensures that the sample mean is an unbiased and consistent estimator of the population mean, making it a suitable basis for confidence interval construction.
• Analyze how confidence intervals are used in hypothesis testing to make decisions about the null and alternative hypotheses.
  • Confidence intervals and hypothesis testing are closely related concepts in statistical inference. Confidence intervals provide a range of plausible values for the population parameter, while hypothesis testing involves making a decision about the null hypothesis based on the sample data. If the hypothesized value for the population parameter, as specified in the null hypothesis, falls outside the calculated confidence interval, it suggests that the null hypothesis is unlikely to be true, and it can be rejected. Conversely, if the hypothesized value is within the confidence interval, the null hypothesis is not rejected. The level of confidence used in the interval estimation directly affects the probability of making a Type I error (rejecting the null hypothesis when it is true) or a Type II error (failing to reject the null hypothesis when it is false) in the hypothesis testing process.

Related terms

Sampling Distribution: The probability distribution of a statistic, such as the sample mean or sample proportion, calculated from all possible samples of the same size drawn from a population.

Standard Error: The standard deviation of the sampling distribution of a statistic, such as the sample mean or sample proportion, which measures the variability of the statistic across different samples.

Margin of Error: The range of values above and below the sample statistic that defines the confidence interval, which is determined by the level of confidence and the standard error.