In Exercises 11 and 12, find the P-value for the hypothesis test with the standardized test statistic z. Decide whether to reject H0 for the level of significance α.


Two-tailed test, z = 2.57, α = 0.10

Verified step by step guidance

1. Step 1: Understand the problem. This is a two-tailed hypothesis test where the standardized test statistic z = 2.57, and the level of significance α = 0.10. The goal is to find the P-value and decide whether to reject the null hypothesis H₀.
2. Step 2: Recall that for a two-tailed test, the P-value is calculated as the area in both tails of the standard normal distribution beyond the absolute value of the test statistic z. Mathematically, P-value = 2 × P(Z > |z|).
3. Step 3: Use the standard normal distribution table (or a statistical software) to find the probability P(Z > 2.57). This represents the area to the right of z = 2.57 under the standard normal curve.
4. Step 4: Multiply the result from Step 3 by 2 to account for both tails of the distribution, as this is a two-tailed test. This gives the total P-value.
5. Step 5: Compare the P-value obtained in Step 4 with the level of significance α = 0.10. If the P-value is less than α, reject the null hypothesis H₀; otherwise, fail to reject H₀.

Key Concepts

P-value

The P-value is a statistical measure that helps determine the significance of results in hypothesis testing. It represents the probability of obtaining a test statistic as extreme as, or more extreme than, the observed value under the null hypothesis. A smaller P-value indicates stronger evidence against the null hypothesis, leading researchers to consider rejecting it.

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about population parameters based on sample data. It involves formulating a null hypothesis (H0) and an alternative hypothesis (H1), then using sample data to calculate a test statistic. The outcome determines whether there is enough evidence to reject H0 in favor of H1 at a specified significance level (α).

Significance Level (α)

The significance level, denoted as α, is the threshold for determining whether to reject the null hypothesis in hypothesis testing. It represents the probability of making a Type I error, which occurs when the null hypothesis is incorrectly rejected. Common values for α are 0.05, 0.01, and 0.10, with lower values indicating a stricter criterion for significance.