Confidence intervals and Tests using the t- distribution

The t-distribution is a probability distribution that is used to estimate population parameters when the sample size is small and/or when the population variance is not known.
The shape of the t-distribution is similar to that of the normal distribution, but its tails are thicker. This means that it is more tolerant of outliers.
The t-distribution has a single parameter known as the degrees of freedom (df), which influences the exact shape of the distribution. The df is equal to the sample size minus 1.
Confidence intervals using the t-distribution are used to estimate the range in which we are confident the true population parameter (e.g. mean) lies, given a certain level of confidence (e.g. 95%).
Constructing a confidence interval involves: determining the sample mean, determining the t critical value (based on the desired level of confidence and the df), determining the standard error of the mean (sample standard deviation divided by the square root of n), and finally calculating the margin of error (t critical value multiplied by the standard error).
Hypothesis testing using the t-distribution involves comparing sample data to the null hypothesis, which is an assumed value of the population parameter.
To perform a t-test, the test statistic (the difference between the sample mean and the hypothesised population mean divided by the standard error) is calculated and compared to a critical value determined from the t-distribution.
If the absolute value of the test statistic is greater than the critical value, the null hypothesis is rejected. If it is less, the null hypothesis is not rejected.
Types of t-tests include the one-sample t-test (used when comparing a sample mean to a known population mean), the paired t-test (used when comparing two related sample means), and the two-sample t-test (used when comparing two independent sample means).
It is important to check the assumptions of the t-test: the data should come from a random sample, the population should be normally distributed, and the population variances should be equal (for a two-sample t-test). Violations of these assumptions can lead to misleading results.