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In hypothesis testing, the test statistic is a significant metric used to decide whether a hypothesis should be accepted or rejected. It's like a benchmark number that helps to compare your sample data against a null hypothesis. For instance, in the original exercise, the test statistic is calculated using the formula: \(Z = \frac{\bar{x} - \mu}{s / \sqrt{n}}\). Here:

• \(\bar{x}\) is the sample mean, which is 112.3 in this case.
• \(\mu\) stands for the population mean under the null hypothesis, which is 120.
• \(s\) represents the sample standard deviation, given as 18.4.
• \(n\) is the number of observations in the sample, which is 100.

Plug these numbers into the formula, and you'll find the test statistic, commonly represented by \(Z\) in a Z-test.
This Z-value helps in determining how far or close the sample mean is to the population mean as stated in the null hypothesis, under the assumption of standard normal distribution. After you compute this value, it's compared against the critical value to proceed to the decision-making step of the hypothesis testing.

The critical value is a threshold that the test statistic is compared against to decide the fate of the null hypothesis. Imagine it as a boundary line. If the test statistic crosses this line, the null hypothesis may be rejected. In the given exercise, the critical value is set based on the significance level and the nature of the test.
Here are some important aspects:

• For this lower-tailed test (because \(H_{a}: \mu < 120\)), the critical value is retrieved from the standard normal distribution table.
• At a significance level of 5%, the critical value is \(-1.645\). This means that if our test statistic is less than \(-1.645\), it falls in the critical region, suggesting strong evidence against the null hypothesis.

Remember, the critical value is dependent on the chosen significance level and determines where the critical region of rejection lies. Knowing how to determine and use the critical value is crucial in hypothesis testing as it guides the decision to accept or reject the null.

The significance level, often denoted by \(\alpha\), is a vital concept in hypothesis testing. It indicates the probability of rejecting a true null hypothesis, often seen as the risk of making a Type I error. It sets the standard for what is considered statistically significant.
In hypothesis testing:

• Common significance levels are 0.05, 0.01, and 0.10, with 0.05 being highly common.
• A 5% significance level implies there is a 5% risk of concluding that a difference exists when there is no actual difference.
• Determining the significance level is important before conducting the test as it directly influences the critical value. As seen in the original exercise, a 5% significance level corresponds to a critical value of \(-1.645\) for a one-tailed test.

By setting the significance level, we establish the confidence with which claims regarding the null hypothesis can be made. A lower significance level demands stronger evidence to reject the null hypothesis, offering an extra layer of confidence in decision-making during hypothesis testing.