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In a left-tailed test, we investigate whether a parameter is smaller than a specified value. This type of test is used when our alternative hypothesis indicates a decrease. The hypothesis setup looks like this:

• Null Hypothesis (\(H_{0}\)): suggests the parameter is equal to or greater than a certain value (e.g., \(H_{0}: \mu \geq 75\)).
• Alternative Hypothesis (\(H_{1}\)): suggests the parameter is less than this value (e.g., \(H_{1}: \mu < 75\)).

For instance, consider a case where a company claims their widget weighs at least 75 grams. If we need to verify this claim by checking if the widget weighs less, we perform a left-tailed test. In such tests, we focus on the lower tail of the probability distribution. This means rejecting \(H_{0}\) if we find that the test statistic falls into the critical region on the lower end.

A right-tailed test is designed to determine if a parameter is greater than a specified value. We use this type of test when our alternative hypothesis suggests an increase. Here’s what the hypotheses look like:

• Null Hypothesis (\(H_{0}\)): indicates the parameter is equal to or less than a certain value (e.g., \(H_{0}: \mu = 45\)).
• Alternative Hypothesis (\(H_{1}\)): states that the parameter is greater than this value (e.g., \(H_{1}: \mu > 45\)).

Imagine a scenario where a machine is supposed to fill at least 45 fluid ounces into bottles. To check if the machine is over-filling, you'd use a right-tailed test. This allows us to look at the "right end" or the upper tail of the distribution. If our test statistic exceeds the critical value on this upper end, we reject the null hypothesis.

A two-tailed test is applicable when we want to test if a parameter is significantly different from a certain value, either greater or smaller. This type of test captures any kind of deviation from the specified value, so the alternative hypothesis doesn't lean in a single direction. Here's how the hypotheses appear:

• Null Hypothesis (\(H_{0}\)): the parameter equals a given value (e.g., \(H_{0}: \mu = 23\)).
• Alternative Hypothesis (\(H_{1}\)): the parameter is not equal to this value (e.g., \(H_{1}: \mu eq 23\)).

For example, when a product's sales target is exactly 23 units per day, and you need to determine if actual sales either exceed or fall short of this number, a two-tailed test is appropriate. It examines both tails of the distribution, identifying any statistically significant deviations from the expectation. Thus, it broadens the scope of the analysis compared to one-tailed tests, enabling a more comprehensive evaluation.