Explain why using the t statistic may be an appropriate alternative to using a z-score (use the concept of estimated standard error to justify your answer).
When comparing two groups or determining the significance of a statistic, both the t statistic and the z-score can be used. The t statistic is preferred over the z-score in certain scenarios due to its reliance on the estimated standard error.
The estimated standard error accounts for the variability in the sample data and is used to estimate the standard deviation of the population. In situations where the population standard deviation is unknown, the sample standard deviation is used, which introduces some uncertainty.
The t statistic takes into account this uncertainty by adjusting for the smaller sample size compared to the entire population. This adjustment is made through the degrees of freedom, which is calculated as the sample size minus one. As the degrees of freedom decrease, the t distribution becomes wider and flatter, accounting for the increased uncertainty.
On the other hand, the z-score assumes that the population standard deviation is known, which is rarely the case in practice. The z-score relies on the population parameters rather than the sample statistics. Consequently, using the z-score without knowledge of the population standard deviation may introduce bias and lead to incorrect conclusions.
Therefore, when the population standard deviation is unknown and estimated using the sample standard deviation, the t statistic should be used instead of the z-score. The t statistic properly accounts for the uncertainty in the estimated standard error, providing a more accurate measure of significance in such situations.