Problem 6 - A Modern Introduction to Probability and Statistics: Understanding Why and How

Question

To test a certain null hypothesis Ho one uses a test statistic \(T\) with a continnous sampling distribution. One agrees that \(H_{0}\) is rejected if one observes a value \(t\) of the test statistic for which (under \(H_{0}\) ) the right tail probability P \((T \geq t)\) is smaller than or equal to \(0.05\). Given below are different values \(t\) and a corresponding left or right tail probability (under \(H_{0}\) ). Specify for each case what the \(p\)-value is, if possible, and whether we should reject \(H_{0-}\) a. \(t=2.34\) and \(P(T \geq 2.34)=0.23\). b. \(t=2.34\) and \(P(T \leq 2.34)=0.23\). c. \(t=0.03\) and \(P(T \geq 0.03)=0.968 .\) d. \(t=1.07\) and \(P(T \leq 1.07)=0.981\). e. \(t=1.07\) and \(P(T \leq 2.34)=0.01\). f. \(t=2.34\) and \(P(T \leq 1.07)=0.981\). g. \(t=2.34\) and \(P(T \leq 1.07)=0.800 .\)

Step-by-Step Solution

Answer

Reject \(H_0\) for case d. Do not reject \(H_0\) for cases a, b, c. Cases e, f, g are inconclusive.

Step 1: Understanding the problem statement

We need to determine the p-value for each scenario and decide whether to reject the null hypothesis, \(H_0\), based on the rejection criterion \(P(T \geq t) \leq 0.05\).

Step 2: Case a: Right Tail Probability Provided

Given: \(t=2.34\), \(P(T \geq 2.34) = 0.23\). This is already the p-value. Since \(0.23 > 0.05\), we do not reject \(H_0\).

Step 3: Case b: Left Tail Probability Provided

Given: \(t=2.34\), \(P(T \leq 2.34) = 0.23\). Calculate right tail probability: \(P(T \geq 2.34) = 1 - 0.23 = 0.77\). Since \(0.77 > 0.05\), we do not reject \(H_0\).

Step 4: Case c: Right Tail Probability Provided

Given: \(t=0.03\), \(P(T \geq 0.03) = 0.968\). This is the p-value. Since \(0.968 > 0.05\), we do not reject \(H_0\).

Step 5: Case d: Left Tail Probability Provided

Given: \(t=1.07\), \(P(T \leq 1.07) = 0.981\). Calculate right tail probability: \(P(T \geq 1.07) = 1 - 0.981 = 0.019\). Since \(0.019 \leq 0.05\), we reject \(H_0\).

Step 6: Case e: Different than Provided \(t\)

Given: \(t=1.07\), but provided \(P(T \leq 2.34) = 0.01\). The provided probability isn't useful since it doesn't match the \(t\) value given for this test. Hence, inconclusive based on this information.

Step 7: Case f: Mismatched Information

Given: \(t=2.34\), but provided probability \(P(T \leq 1.07) = 0.981\) doesn't apply since it doesn't relate to the \(t\) value used. Inconclusive.

Step 8: Case g: Mismatched Information

Given: \(t=2.34\), but provided probability \(P(T \leq 1.07) = 0.800\) again is mismatched with the \(t\) value. Considered inconclusive without more information.

Key Concepts

Null Hypothesis

The Null Hypothesis, often denoted as \( H_0 \), is a statement used in statistical hypothesis testing. It is the assumption that there is no significant effect or difference in the context of the experiment or study. Essentially, it serves as the default or baseline proposition that researchers aim to test. In the context of the given exercise, \( H_0 \) assumes that any observed effect or difference in test statistics is due to chance rather than some underlying effect or condition. If evidence strongly contradicts \( H_0 \), researchers may reject it in favor of an alternative hypothesis (\( H_1 \)). The null hypothesis is central to hypothesis testing because it provides the foundation against which outcomes are compared in statistical analysis.

Test Statistic

The test statistic is a standard measure used to decide whether to reject the null hypothesis. It is computed from sample data and follows a known distribution under the null hypothesis. In our exercise, it is denoted as \( T \), and specific values, such as \( t=2.34 \) and \( t=1.07 \), are used to evaluate the strength of evidence against \( H_0 \).
The value of the test statistic helps determine the p-value, which in turn informs our decision on the null hypothesis. A higher test statistic often indicates a greater deviation from the null hypothesis. However, understanding the specific context and distribution of the test statistic is crucial since it directly affects which tail probability will be used (left or right) to assess significance.

Right Tail Probability

Right tail probability refers to the probability of observing a test statistic as extreme or more extreme in the positive direction than the one observed, given that the null hypothesis is true. In simpler terms, if you imagine a bell curve representing the distribution of the test statistic, the right tail probability is the area under the curve to the right of the observed statistic.
In hypothesis testing, if this probability (the p-value) is smaller than a predetermined threshold, such as \( 0.05 \), it suggests that the observed result is unlikely under the null hypothesis and may lead to its rejection. This is particularly relevant in scenarios where a right-tailed test is appropriate, typically when testing for statistical significance in increases or positive effects.

Rejection Criterion

The rejection criterion in hypothesis testing is the rule or condition under which the null hypothesis is rejected. It is typically based on the p-value, which represents the probability of obtaining the observed data or something more extreme given that the null hypothesis is true. For many tests, the threshold for significance is set at \( 0.05 \).
In our example, if \( P(T \geq t) \leq 0.05 \), then \( H_0 \) is rejected, indicating a statistically significant result. If the p-value exceeds \( 0.05 \), then \( H_0 \) is not rejected, suggesting insufficient evidence against the null hypothesis. This criterion ensures that researchers have a consistent method for making decisions about hypotheses based on statistical evidence.