91Ó°ÊÓ

In statistics, a two-tailed test is used when we want to determine if a sample mean is significantly greater than or lesser than a hypothesized population mean. This test is called "two-tailed" because the critical region for rejection is in two tails of the sampling distribution.

Here's how it works: Imagine a bell curve that represents the distribution of sample means. When conducting a two-tailed test, we divide our significance level, denoted by \( \alpha \), into two equal parts. For example, if \( \alpha = 0.01 \), each tail will have an area of \( 0.005 \).

• On the left tail, this \( 0.005 \) represents extremely low values.
• On the right tail, another \( 0.005 \) represents extremely high values.

These are called critical values, and if our sample mean falls beyond these values, we reject the null hypothesis, suggesting it’s significantly different from the population mean. This ensures a balanced approach, accounting for deviations on both ends of the scale.

A left-tailed test is used when we aim to determine if a sample mean is significantly less than the population mean. This test is called "left-tailed" because the critical region lies entirely in the left tail of the distribution.

Consider the following steps: You start with your significance level \( \alpha \). For instance, if \( \alpha = 0.005 \), the entire \( \alpha \) is placed in the left tail of the sampling distribution.

• This means we're checking for values that are extremely low relative to the mean.
• The critical value, often denoted as \( Z_{0.005} \), marks the threshold for rejection.

If the test statistic for your sample is below this critical value, you reject the null hypothesis.

This test focuses solely on the possibility of a decrease, making it useful when a lower mean is hypothesized.

When you expect a sample mean to be significantly greater than a hypothesized population mean, you would use a right-tailed test. As expected, the "right-tailed" designation indicates the entire critical region is in the right tail of the distribution.

Here's the process for this test: With your significance level \( \alpha \), say \( \alpha = 0.025 \), the full \( \alpha \) is placed on the right side.

• This area covers the possibility of values being excessively high compared to the mean.
• Here, the critical value \( Z_{0.025} \) acts as the cutoff for rejection.

If the sample test statistic exceeds this critical threshold, then the null hypothesis is rejected.

This test is optimal when exploring possibilities for an increase in the mean.

Critical values play the role of decision-makers in hypothesis testing. They are the cutoff points that help us decide whether to reject the null hypothesis.

Each hypothesis test has its own critical values, determined based on the significance level \( \alpha \), the type of test being conducted, and the sample size.

• For two-tailed tests, there are two critical values.
• For left-tailed and right-tailed tests, there is a single critical value.

These values are often obtained from statistical tables such as z-tables or t-tables depending on the distribution required. In a z-test, these critical values are denoted as \( Z_{\alpha} \) for right-tailed tests, \( Z_{-\alpha} \) for left-tailed tests, and both for two-tailed tests.

When your test statistic lies beyond these critical values, it means there is significant evidence against the null hypothesis, leading to its rejection.