When the population standard deviation is unknown, a different method must be used to perform a hypothesis test for the mean of a single sample. The most commonly used method is the Student’s t-test. The t-test is used when the sample size is small or the population standard deviation is unknown.

The t-test is based on the t-distribution, which is a probability distribution that is similar to the standard normal distribution (also known as the z-distribution), but has more spread and “fatter tails.” This means that the t-distribution is more forgiving of small sample sizes and unknown population standard deviations.

In a one-sample t-test, the null hypothesis is that the population mean is equal to a specified value (often denoted as μ0), and the alternative hypothesis is that the population mean is not equal to the specified value.

The test statistic used in a one-sample t-test is the t-value, which is calculated using the following formula:

t = (x̄ – μ0) / (s / √n)

where x̄ is the sample mean, μ0 is the specified population mean, s is the sample standard deviation, and n is the sample size.

The t-value is then used to find the p-value, which is the probability of getting a t-value as extreme or more extreme than the calculated t-value, assuming that the null hypothesis is true. The p-value is found by looking up the t-value in a t-distribution table or using a calculator that uses the t-distribution.

The decision about whether to reject or fail to reject the null hypothesis is based on the p-value and the significance level (alpha) set by the researcher. If the p-value is less than the significance level, then the null hypothesis is rejected, and it is concluded that there is evidence to suggest that the population mean is different from the specified value. If the p-value is greater than the significance level, then the null hypothesis is not rejected, and it is concluded that there is not enough evidence to suggest that the population mean is different from the specified value.

It is important to note that the sample size has a big role in this test, the larger the sample size, the more power the test has to detect the difference between the sample mean and the population mean, which means that the t-value will be larger and the p-value will be smaller.

When the sample size is small (typically less than 30), the t-distribution should be used with caution. In these cases, the sample standard deviation should be used as an estimate of the population standard deviation. And this could make the t-value larger, and the p-value smaller, which make the test more conservative.

Another important thing to consider when using t-test is that the data should be approximately normally distributed. If the data is heavily skewed or has outliers, it may be more appropriate to use a non-parametric test such as the Wilcoxon signed-rank test or the Mann-Whitney U test.

In summary, the t-test is a commonly used method for performing a hypothesis test for the mean of a single sample when the population standard deviation is unknown. It is based on the t-distribution, which is more forgiving of small sample sizes and unknown population standard deviations. The t-test involves calculating a t-value and a p-value, and using these values to make a decision about whether to reject or fail to reject the null hypothesis. It’s essential to consider the sample size and the normality of the data when using t-test.