Explain why using the t statistic may be an appropriate alternative to using a z-score
Usually you use z-tests when sample sizes are large (n is greater than
or equal to 30) whether or not you know the population standard deviation.
If you do not know the population standard deviation and have a small
sample (n < 30), then you can use t-tests.
The t statistic is often used as an alternative to the z-score when dealing with small sample sizes or when the population standard deviation is unknown. Here's why it may be appropriate to use the t statistic:
1. Sample size: The z-score assumes that you have a large sample size (typically greater than 30 observations) because it relies on the central limit theorem. However, if you have a small sample size, the t statistic is more appropriate. The t statistic takes into account the degrees of freedom, which is calculated based on the sample size and accounts for the inherent variability in small samples.
2. Unknown population standard deviation: The z-score requires knowledge of the population standard deviation, which is often unknown in real-world scenarios. In such cases, the sample standard deviation is used instead. The t statistic is calculated by dividing the difference between the sample mean and the hypothesized population mean by the standard error of the sample mean, which is calculated using the sample standard deviation.
To calculate the t statistic:
1. Calculate the sample mean (x̄) and sample standard deviation (s) from your data.
2. Specify your null and alternative hypotheses. Typically, the null hypothesis assumes no difference between the sample mean and the hypothesized population mean, while the alternative hypothesis asserts a difference.
3. Calculate the standard error of the sample mean using the formula: SE = s / √n, where s is the sample standard deviation and n is the sample size.
4. Calculate the t statistic using the formula: t = (x̄ - μ) / SE, where x̄ is the sample mean, μ is the hypothesized population mean, and SE is the standard error of the sample mean.
5. Find the critical t-value or p-value associated with the desired level of significance (e.g., 0.05). Compare the calculated t statistic to this critical value to determine significance.
Using the t statistic in scenarios where the sample size is small or the population standard deviation is unknown ensures more accurate and reliable inference about the population parameters.