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In hypothesis testing, the critical value is a threshold used to decide whether to reject the null hypothesis. It marks the boundary of the rejection region, where the test statistic must fall for the null hypothesis to be rejected. The critical value depends on the chosen significance level, denoted by \( \alpha \), and the type of test being conducted. For a two-tailed test, the significance level is split between both tails, while for a one-tailed test, it is concentrated in one tail.
Understanding critical values is crucial, as they directly impact the confidence we can have in the decision made. For example, if the significance level is \( \alpha = 0.05 \), the critical value is determined such that there is a 5% chance of wrongly rejecting the true null hypothesis.

• Critical values for the z-distribution can be found in the standard normal distribution table.
• Critical values for the t-distribution depend on the degrees of freedom and can be found in t-distribution tables.

By knowing the critical value, we can compare it to our test statistic and make an informed decision about the null hypothesis.

A two-tailed test is employed when the alternate hypothesis indicates that the parameter could be either less than or greater than the null hypothesis value. The critical regions are located at both ends of the distribution. For instance, when testing \( \mathrm{H}_0: \mu = 105 \) against \( \mathrm{H}_A: \mu eq 105 \), we are considering both possibilities: \( \mu < 105 \) and \( \mu > 105 \).
Two-tailed tests require splitting the significance level \( \alpha \) between the two tails of the distribution. This means for a significance level of 0.05, each tail gets \( \alpha/2 = 0.025 \).

• Two-tailed tests are typically used when there is no specific direction of interest.
• It is important to look for an alternate hypothesis that includes a non-equal sign \((eq)\).

Because of the split, the critical values are less extreme compared to one-tailed tests with the same \( \alpha \), making them generally more conservative in rejecting the null hypothesis.

A right-tailed test is conducted when the alternate hypothesis specifies a parameter greater than the null hypothesis value. This is indicated by a 鈥済reater than鈥� sign \((>)\) in the alternate hypothesis, such as in \( \mathrm{H}_0: p = 0.05 \) versus \( \mathrm{H}_A: p > 0.05 \).
In this kind of test, the entire significance level \( \alpha \) is concentrated in the right tail of the distribution, where we look for values that significantly exceed expectations under the null hypothesis.

• A right-tailed test is appropriate when testing for an increase or an improvement.
• The critical value is found by identifying the point where the right tail area equals \( \alpha \), often using a z-table.

The decision rule is straightforward: if the test statistic is greater than the critical value, we reject the null hypothesis, supporting the claim that the true parameter is indeed greater.

A left-tailed test investigates if the parameter is significantly less than the null hypothesis value, indicated by a 鈥渓ess than鈥� sign \((<)\) in the alternate hypothesis, like in the case of \( \mathrm{H}_0: p = 0.5 \) versus \( \mathrm{H}_A: p < 0.5 \).
Here, the significance level \( \alpha \) is placed entirely in the left tail of the distribution. This setup aims to detect values that fall significantly below what the null hypothesis predicts.

• Left-tailed tests are applicable when there is a suspicion of a decrease or decline.
• The critical value for a left-tailed test is obtained when the area to the left equals \( \alpha \), using either a z-table or a t-table.

If the test statistic is less than the critical value, we reject the null hypothesis, suggesting that the parameter is lower than what the null hypothesis states.