Intervalo de Confiança para a MÉDIA (μ) com variância populacional desconhecida

Summary

This video tutorial explains how to calculate a confidence interval for a population mean when the population variance is unknown. It covers key concepts such as sample mean, sample standard deviation, and the use of the Student’s t-distribution. The tutorial outlines the formula for calculating the confidence interval and demonstrates its application through two practical examples: one involving the weight of candies and the other concerning crime statistics. Viewers learn how to interpret the confidence interval, understand the margin of error, and how factors like sample size and confidence level affect the result.

Takeaways

• 😀 Confidence intervals are used to estimate the population mean when the population variance is unknown.
• 😀 A random sample should be selected, and the population should ideally be normally distributed (or large enough for the Central Limit Theorem to apply).
• 😀 When the population variance is unknown, the sample standard deviation (S) is used as an estimator for the population standard deviation (σ).
• 😀 The formula for a confidence interval when the variance is unknown is: (x̄ - tα/2 * S/√n, x̄ + tα/2 * S/√n).
• 😀 The critical value tα/2 is taken from the t-distribution with n-1 degrees of freedom, based on the desired confidence level.
• 😀 The margin of error is calculated as: E = tα/2 * S/√n.
• 😀 Larger sample sizes reduce the margin of error, while larger sample standard deviations increase the margin of error.
• 😀 Increasing the confidence level (e.g., from 95% to 99%) increases the critical value (tα/2), thus increasing the margin of error.
• 😀 For a 95% confidence level, the critical value (tα/2) for a sample size of 29 is 2.048, and for a 99% confidence level with 20 samples, it is 2.861.
• 😀 In practice, confidence intervals can be calculated using the sample mean, sample standard deviation, and the sample size, along with the appropriate t-distribution value.
• 😀 Example 1: In a bombon production case, the confidence interval for the mean weight of the bombons is between 526.39g and 541.66g at a 95% confidence level.
• 😀 Example 2: In a crime statistics case, the confidence interval for the mean number of crimes per municipality is between 21.76 and 152.24 at a 99% confidence level.

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