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A two-tailed test is a type of hypothesis test used when we want to determine if there is a significant difference between sample data and a null hypothesis, considering potential deviations in both directions of a distribution. In simpler terms, a two-tailed test checks for evidence that the true parameter is either greater than or less than a hypothesized value. This is particularly useful when you do not have a directional expectation about the change.

When performing a two-tailed test, the null hypothesis is typically expressed as the parameter being equal to a certain value, and the alternative hypothesis suggests that the parameter is not equal to that value. Thus, the test can detect deviations on both sides of the hypothesized parameter.

Key aspects of a two-tailed test include:

• Both tails of the probability distribution are considered.
• The critical values are set such that they cover both ends of the distribution.
• If the test yields results far from the center value, it points to rejecting the null hypothesis.

The total P-value in a two-tailed test represents the probability of observing a test statistic as extreme, or more extreme, than the one calculated in either direction. Therefore, when given a P-value in a two-tailed context, like 0.0134 in our exercise, it is critical to understand that this value is split equally between the two tails, making it necessary to divide by two to find the P-value for either one-side tests.

Hypothesis testing is a fundamental statistical method used to make decisions or infer conclusions about a population based on sample data. By using hypothesis testing, we can determine the likelihood that an observed effect or relationship exists within a given population.

Here are the steps involved in hypothesis testing:

• Formulate the null hypothesis (H_0): This is usually a statement of no effect or no difference, asserting that any deviation observed is due to random chance.
• Propose an alternative hypothesis (H_a): This is what you want to prove or demonstrate, a statement indicating the presence of an effect or difference.
• Calculate a test statistic: This is a standardized value derived from the sample data which helps in deciding whether to reject the null hypothesis.
• Determine the P-value: This statistic indicates the probability of observing the effect in the sample data if the null hypothesis is true.
• Make a decision: By comparing the P-value with a pre-determined significance level (usually 0.05), you can decide whether or not to reject the null hypothesis.

The role of the P-value is crucial in hypothesis testing, as it quantifies the strength of evidence against the null hypothesis. The smaller the P-value, the stronger the evidence against the null hypothesis, suggesting that an alternative hypothesis may better explain the observed data.

A left-tailed test, also known as a one-tailed test, investigates whether a sample parameter is significantly less than a specified value. Here, we focus on the left side of the probability distribution, searching for smaller than expected outcomes.

In practice, a left-tailed test includes:

• Direction: The hypothesis is directional, predicting a deviation in one specific direction (less than).
• Null and Alternative Hypotheses: The null hypothesis might state that a parameter is greater than or equal to a certain value, whereas the alternative will suggest it is less than that value.
• Significance Level: The entire significance level is allocated to the left tail of the distribution.

Unlike the two-tailed test, a left-tailed test is concerned with detecting evidence only on the lower end of the distribution. In our example, to compute the left-tailed P-value, you take the total P-value from a two-tailed test and divide it by two, as the original P-value is equally distributed among both tails. Thus, for a P-value of 0.0134 from a two-tailed test, the left-tailed equivalent is 0.0067, highlighting the smaller and more specific focus of this test.