Finding P-Values
In Exercises 13–16, do the following:

i. Identify the hypothesis test as being two-tailed, left-tailed, or right-tailed.
ii. Find the P-value. (See Figure 8-3.)
iii. Using a significance level of α = 0.05 should we reject H0 or should we fail to reject H0?

The test statistic of z = -1.60 is obtained when testing the claim that p ≠ 0.455.

Verified step by step guidance

Step 1: Identify the type of hypothesis test. Since the claim is that p ≠ 0.455, this is a two-tailed test because the alternative hypothesis (H1) involves a 'not equal to' condition, which tests for deviations in both directions (greater than or less than).

Step 2: Determine the P-value. For a two-tailed test, the P-value is calculated as the sum of the probabilities in both tails of the standard normal distribution beyond the absolute value of the test statistic z = -1.60. Use the standard normal table or a statistical software to find the area to the left of z = -1.60, then double it to account for both tails.

Step 3: Compare the P-value to the significance level α = 0.05. If the P-value is less than or equal to 0.05, we reject the null hypothesis (H0). If the P-value is greater than 0.05, we fail to reject H0.

Step 4: Interpret the result. Based on the comparison in Step 3, decide whether there is sufficient evidence to support the claim that p ≠ 0.455 or if the evidence is insufficient to reject the null hypothesis.

Step 5: Conclude the hypothesis test. Clearly state whether H0 is rejected or not and provide a summary of the findings in the context of the problem, ensuring the decision aligns with the significance level and the calculated P-value.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population parameter based on sample data. It involves formulating two competing hypotheses: the null hypothesis (H0), which represents no effect or no difference, and the alternative hypothesis (H1), which represents the effect or difference. The goal is to determine whether there is enough evidence in the sample to reject H0 in favor of H1.

P-Value

The P-value is a measure that helps determine the strength of the evidence against the null hypothesis. It represents the probability of obtaining a test statistic as extreme as, or more extreme than, the observed value under the assumption that H0 is true. A smaller P-value indicates stronger evidence against H0, and it is compared to a predetermined significance level (α) to decide whether to reject H0.

One-Tailed vs. Two-Tailed Tests

In hypothesis testing, tests can be classified as one-tailed or two-tailed based on the direction of the alternative hypothesis. A one-tailed test assesses the possibility of the relationship in one direction (either greater than or less than), while a two-tailed test evaluates both directions (not equal to). The choice between these tests affects the critical values and the interpretation of the P-value.

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