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What is a Confidence Interval?

A confidence interval is a statistical range or interval that estimates the true value of a population parameter, such as the mean, proportion, or difference between means. It provides a measure of uncertainty surrounding the estimate by indicating the range of values within which the true parameter value is likely to lie, along with a level of confidence associated with this estimation.

Explanation:

Confidence intervals are constructed based on sample data and are influenced by factors such as sample size, variability of the data, and the chosen confidence level. The confidence level typically expressed as a percentage (e.g., 95% confidence level), represents the probability that the interval contains the true population parameter if the sampling process is repeated multiple times.

The formula for constructing a confidence interval varies depending on the type of parameter being estimated (e.g., mean, proportion) and the distribution of the data (e.g., normal distribution, t-distribution). However, in general, confidence intervals are calculated using point estimates (such as sample means or proportions) along with standard errors, critical values from appropriate probability distributions, and a margin of error.

Example:

Suppose we want to estimate the average height of students in a university. We collected a random sample of 100 students and found that the average height in the sample is 65 inches, with a standard deviation of 3 inches.

To construct a 95% confidence interval for the population mean height, we use the formula:

Confidence Interval=Sample Mean±(Critical Value×Standard Deviation/V-Sample Size)

Using a normal distribution and a 95% confidence level (which corresponds to a critical value of approximately 1.96), we have:

Confidence Interval = 65 ± (1.96 × 3V100)

Confidence Interval = 65 ± (1.96 × 0.3)

Confidence Interval = 65 ± 0.588

Thus, the 95% confidence interval for the population mean height is approximately (64.412,65.588) (64.412,65.588) inches. This means that we are 95% confident that the true average height of all students in the university falls within this range.

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1. Confidence Intervals for Means: Understanding how to calculate confidence intervals for population means is essential in statistics. We provide help with determining confidence intervals for means based on sample data.
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