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The t distribution accounts for the additional uncertainty introduced by using the sample standard deviation as an estimate for the unknown population standard deviation. The t distribution has thicker tails than the normal distribution, producing wider confidence intervals that better reflect this uncertainty. This approach is more accurate, especially with smaller sample sizes, compared to estimating population mean with known standard deviation methods.

The shape of the Student's t distribution depends on degrees of freedom, calculated as n minus one. As sample size increases, degrees of freedom increase, and the t distribution becomes increasingly similar to the standard normal distribution. With larger samples, the distribution becomes narrower in the tails and taller in the center, approaching the normal curve's shape.

The critical t value, denoted as tα/2, is not constant and changes with sample size through degrees of freedom. Degrees of freedom equal the sample size minus one. As degrees of freedom increase, the critical t value decreases and approaches the corresponding z value, reflecting reduced uncertainty with larger samples.

Use the Student's t distribution whenever the sample standard deviation is used as an estimate for the unknown population standard deviation. The underlying population should be approximately normally distributed, and the sample size should exceed 30. Modern practice favors using the t distribution for all sample sizes when the population standard deviation is unknown.

When the population standard deviation is unknown, the margin of error uses the critical t value and sample standard deviation instead of the z value and population standard deviation. The t value is generally greater than the corresponding z value, producing a wider margin of error and broader confidence interval to account for the additional uncertainty from estimating the population parameter.

William Gosset, a statistician at the Guinness brewery, encountered problems when calculating confidence intervals with small sample sizes. Simply replacing the population standard deviation with the sample standard deviation produced inaccurate results. He discovered that the actual distribution depends on sample size, leading him to develop what became known as the Student's t distribution, published under his pen name.

The underlying population should be approximately normally distributed with unknown mean and unknown standard deviation. Random sampling is required to obtain the sample. The population size is generally not relevant unless extremely small. These assumptions ensure that the t distribution accurately models the sampling distribution and produces reliable confidence interval for estimating population mean.