To determine the critical value for a one-tailed hypothesis test comparing two population means (H0: μ1 ≤ μ2) at a 5% significance level, with both sample sizes equal to 13 and assuming equal population standard deviations, you need to use a t-distribution. The degrees of freedom (df) for the test can be calculated as: df = (n1 - 1) + (n2 - 1) df = (13 - 1) + (13 - 1) df = 24 Using a t-distribution table or software, look up the critical t-value corresponding to a one-tailed test with 24 degrees of freedom and a 5% significance level (0.05 in the tail). The exact critical value can vary slightly depending on the source, but it is approximately 1.711. Always round to a suitable number of decimal places as required by the context, or refer to a specific t-distribution table or statistical software output for a precise value.

To determine the critical value for a one-tailed hypothesis test at a 5% significance level using the given information, we first calculate the degrees of freedom as you have rightly done:

df = (n1 - 1) + (n2 - 1) df = (13 - 1) + (13 - 1) df = 12 + 12 df = 24

Since a t-distribution is used and we have calculated 24 degrees of freedom, we look up the critical t-value for a one-tailed test with df=24 at the 5% significance level. For a one-tailed test at this level, we would typically look for the t-value that corresponds to an area of 0.05 in the upper tail of the t-distribution. Consulting a t-table or using statistical software will give us the critical value, which is approximately 1.711 as you mentioned. However, it's crucial to consult your specific t-table or statistical software as values can slightly differ. The critical value tells you the cutoff point at which you would reject the null hypothesis (H0: μ1 ≤ μ2) in favor of the alternative hypothesis (H1: μ1 > μ2) if the test statistic computed from your data exceeds this critical value.

Extra: Understanding the critical value and the t-distribution is important for hypothesis testing. The t-distribution is a probability distribution that is used when the sample size is small (usually less than 30) and the population standard deviation is unknown. It is similar to the normal distribution but has heavier tails, meaning there is a greater chance of obtaining values far from the mean. The critical value from the t-distribution is a boundary that helps us decide whether to accept or reject the null hypothesis. If the computed test statistic from our data is greater than the critical value in a one-tailed test, this means that our test statistic falls into the rejection region, and we have sufficient evidence to reject the null hypothesis in favor of the alternative hypothesis. In the context of comparing two population means, if the test statistic is greater than the critical value, it suggests that there is a significant difference between the two means at the given level of significance, and we conclude that μ1 is likely greater than μ2, assuming the direction specified in the alternative hypothesis.