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In statistics, a t-test is a method used to determine if there is a significant difference between the means of two groups. It is particularly useful when the sample size is small and the standard deviation is unknown, which fits the scenario in our exercise. Here, we use the t-test to compare the mean score of a revised exam to a known average from past years. The test requires calculating a test statistic based on the sample data: \( t = \frac{\bar{x} - \mu}{s/\sqrt{n}} \). In this case, \( \bar{x} \) is the sample mean (14.6), \( \mu \) is the population mean (14.3), \( s \) is the standard deviation (2.4), and \( n \) is the sample size (30). This test statistic helps us understand how the observed data compares to what we would expect under the null hypothesis.

The critical value approach is a traditional method used in hypothesis testing. It involves comparing your calculated test statistic to a critical value obtained from statistical tables. To employ this approach, you need to choose a significance level (such as \(10\%\)), and determine whether the test is one-tailed or two-tailed. Then, based on the degrees of freedom (sample size minus one for the t-test), find the critical values. In our problem, with 29 degrees of freedom and a two-tailed test at a \(10\%\) level, the critical t-values are around \(±1.699\). Finally, check if your test statistic lies within the acceptance region, defined by these critical values. If it does, you fail to reject the null hypothesis.

The p-value approach provides another way to perform hypothesis testing. It refers to the probability of observing a test statistic as extreme as, or more extreme than, the value calculated from your data, assuming the null hypothesis is true. A p-value helps quantify the evidence against the null hypothesis. In our exercise, the p-value is approximately 0.50. You compare this to your chosen level of significance (often denoted as \( \alpha \)). Since the p-value is higher than \( \alpha = 0.10 \), we do not have enough evidence to reject the null hypothesis. This approach is favored for its straightforward interpretation and ability to show the strength of evidence.

The significance level, often denoted by \( \alpha \), is a critical component of hypothesis testing. It represents the probability of rejecting the null hypothesis when it is actually true. In simpler terms, it's the risk you are willing to take for a Type I error—concluding that there is an effect or difference when none exists. In our problem, a \( 10\% \) significance level means you allow a 10% chance of incorrectly rejecting the null hypothesis. This level determines the critical value(s) used in the critical value approach and is a benchmark for comparing your p-value in the p-value approach. Choosing an appropriate significance level is pivotal to balancing the risk of Type I errors with the confidence in your test results.