Graphical Analysis In Exercises 21 and 22, state whether each standardized test statistic z allows you to reject the null hypothesis. Explain your reasoning.


a. z = -1.301
b. z = 1.203
c. z = 1.280
d. z = 1.286

Verified step by step guidance

1. Step 1: Understand the context of the problem. The standardized test statistic z is used in hypothesis testing to determine whether to reject the null hypothesis. The decision depends on the critical value and the significance level (α). Typically, for a two-tailed test with α = 0.05, the critical z-values are approximately ±1.96.
2. Step 2: Compare each z-value provided in the problem (z = -1.301, z = 1.203, z = 1.280, z = 1.286) to the critical z-value. If the absolute value of z is greater than the critical value (e.g., |z| > 1.96 for α = 0.05), the null hypothesis is rejected. Otherwise, it is not rejected.
3. Step 3: Analyze the graph provided. The graph shows a z-distribution with a critical z-value of z₀ = 1.285, which appears to represent the boundary for rejection in a one-tailed test. This suggests that z-values greater than 1.285 fall into the rejection region.
4. Step 4: For each z-value: (a) z = -1.301 is less than the critical value and does not fall into the rejection region. (b) z = 1.203 is less than the critical value and does not fall into the rejection region. (c) z = 1.280 is less than the critical value and does not fall into the rejection region. (d) z = 1.286 is greater than the critical value and falls into the rejection region.
5. Step 5: Conclude that only z = 1.286 allows you to reject the null hypothesis based on the graph and the critical z-value provided. The reasoning is that it exceeds the critical z-value, indicating statistical significance.

Key Concepts

Standardized Test Statistic (z)

A standardized test statistic, commonly denoted as z, measures how many standard deviations an element is from the mean. In hypothesis testing, the z-value helps determine whether to reject the null hypothesis by comparing it to critical values from the standard normal distribution.

Null Hypothesis (H0)

The null hypothesis (H0) is a statement that there is no effect or no difference, and it serves as the default assumption in hypothesis testing. The goal of statistical tests is to determine whether there is enough evidence to reject H0 in favor of an alternative hypothesis (H1).

Critical Value

A critical value is a threshold that defines the boundary for rejecting the null hypothesis. In the context of a z-test, if the calculated z-value exceeds the critical value (e.g., 1.285 in the provided image), it indicates that the result is statistically significant, leading to the rejection of H0.