Article 5. An introduction to estimation--2: from z to t

Area of Science:

• Statistics
• Inferential Statistics

Background:

• When population standard deviation (sigma) is unknown, statistical inference relies on sample data.
• Sample size is a critical factor in determining the appropriate statistical test for population parameter estimation.

Purpose of the Study:

• To elucidate the application of the z statistic for large sample sizes (n>100) in estimating population mean and proportion.
• To highlight the role of the t statistic as a replacement for the z statistic when dealing with smaller sample sizes and unknown population standard deviation.

Main Methods:

• Utilizing the z statistic for confidence interval estimation of the population mean with large samples, employing the standard error of the mean when sigma is unknown.
• Applying the z statistic for population proportion estimation with large samples.
• Employing the t statistic for estimations involving smaller samples with unknown population standard deviation.

Main Results:

• The z statistic is valid for estimating population mean confidence intervals and proportions when sample size exceeds 100, even if sigma is unknown.
• The t statistic effectively replaces the z statistic for smaller samples, enabling estimations of probabilities related to sample means and confidence intervals for the population mean.

Conclusions:

• The choice between z and t statistics is contingent upon sample size and knowledge of population standard deviation.
• Both z and t statistics are crucial tools for inferential statistics, providing methods to estimate population parameters from sample data.