Understanding Confidence Intervals for Effective Statistical
School
University of Calgary**We aren't endorsed by this school
Course
STAT 423
Subject
Statistics
Date
Dec 10, 2024
Pages
2
Uploaded by BaronSquirrel4760
7. Confidence IntervalsA confidence interval is a range of values used to estimate the true value of a population parameter. It is calculated as:Confidence Interval=θ^±Z⋅σn\text{Confidence Interval} = \hat{\theta} \pm Z \cdot \frac{\sigma}{\sqrt{n}}Confidence Interval=θ^±Z⋅nStatistical inference and hypothesis testing are essential tools in econometrics, statistics, and research. They allow us to estimate population parameters, test theories, and make informed decisions based on sample data. By understanding hypothesis testing steps, errors, and test statistics (Z-test, T-test, Chi-square, F-test), researchers can draw valid conclusions and determine the significance of their findings.Confidence Intervals (CI)A confidence intervalis a statistical tool used to estimate a population parameter, such as the mean or proportion, based on sample data. The interval provides a range of values that is likely to contain the true value of the population parameter. A key feature of a confidence interval is the associated confidence level, usually expressed as a percentage (e.g., 95% or 99%).Formula for Confidence Interval:The confidence interval for a population parameter, such as the population mean, is given by the formula:Confidence Interval=θ^±Z⋅σn\text{Confidence Interval} = \hat{\theta} \pm Z \cdot \frac{\sigma}{\sqrt{n}}Confidence Interval=θ^±Z⋅nWhere:● θ^\hat{\theta}θ^is the sample estimate of the population parameter (e.g., sample mean xˉ\bar{x}xˉ if estimating population mean).● Zis the Z-scorecorresponding to the desired confidence level (e.g., for a 95% confidence level, Z ≈ 1.96). The Z-score represents how many standard deviations the sample estimate is from the population mean under the assumption that the sampling distribution is normal.● σ\sigmaσis the population standard deviation (if known) or the sample standard deviation (if the population standard deviation is unknown).● nis the sample size.
The formula gives a range of values that, with a certain level of confidence, is likely to contain the true population parameter.Example:Imagine you're estimating the average height of adult men in a city, and you sample 100 people. From the sample, the mean height θ^\hat{\theta}θ^ is 175 cm, with a standard deviation σ\sigmaσ of 10 cm. If you want a 95% confidence level (Z ≈ 1.96), the confidence interval would be:Confidence Interval=175±1.96⋅10100=175±1.96⋅1=[173.04,176.96]\text{Confidence Interval} = 175 \pm 1.96 \cdot \frac{10}{\sqrt{100}} = 175 \pm 1.96 \cdot 1 = [173.04, 176.96]Confidence Interval=175±1.96⋅100