A hypothesis test concerning the mean is a statistical test used to determine whether a sample mean is significantly different from a known or hypothesized population mean. The purpose of the test is to evaluate the strength of evidence against the null hypothesis, which states that there is no significant difference between the sample mean and the population mean.

The steps involved in conducting a hypothesis test concerning the mean are as follows:

1. State the null hypothesis (H0) and alternative hypothesis (Ha). The null hypothesis is the hypothesis that there is no significant difference between the sample mean and the population mean. The alternative hypothesis is the hypothesis that there is a significant difference between the sample mean and the population mean.

2. Determine the level of significance (α). This is the probability of rejecting the null hypothesis when it is actually true. The most commonly used level of significance is 0.05.

3. Select the appropriate test statistic. The choice of test statistic depends on the sample size, whether the population standard deviation is known or unknown, and whether the sample is drawn from a normally distributed population.

4. Calculate the test statistic. This involves computing the standard error of the mean, which is a measure of the variability of the sample mean, and using it to calculate the test statistic.

5. Determine the critical value or p-value. The critical value is the value of the test statistic that separates the rejection region from the non-rejection region. The p-value is the probability of obtaining a test statistic as extreme as the one observed, assuming the null hypothesis is true.

6. Make a decision and interpret the results. If the test statistic falls in the rejection region, the null hypothesis is rejected, and it is concluded that there is a significant difference between the sample mean and the population mean. If the test statistic falls in the non-rejection region, the null hypothesis is not rejected, and it is concluded that there is insufficient evidence to support the alternative hypothesis.

In summary, a hypothesis test concerning the mean is a statistical test used to determine whether a sample mean is significantly different from a known or hypothesized population mean. The test involves stating the null and alternative hypotheses, selecting the appropriate test statistic, calculating the test statistic, determining the critical value or p-value, making a decision, and interpreting the results.