Graphical Analysis In Exercises 9–12, state whether each standardized test statistic t allows you to reject the null hypothesis. Explain.


b. t = 0

Verified step by step guidance

1. Step 1: Understand the context of the problem. The standardized test statistic t is used in hypothesis testing to determine whether to reject the null hypothesis. The decision depends on the critical t-value and the significance level (α).
2. Step 2: Analyze the graph provided. The graph shows a t-distribution with a critical t-value of t₀ = -2.086, which marks the rejection region on the left tail of the distribution. The shaded area represents the rejection region.
3. Step 3: Compare the given test statistic t = 0 to the critical t-value. Since t = 0 lies at the center of the distribution and does not fall within the rejection region (shaded area), it does not meet the criteria for rejecting the null hypothesis.
4. Step 4: Explain the reasoning. The null hypothesis is rejected only if the test statistic falls within the rejection region, which is determined by the critical t-value and the significance level. In this case, t = 0 is not extreme enough to reject the null hypothesis.
5. Step 5: Conclude the analysis. Based on the comparison and the graph, the test statistic t = 0 does not allow you to reject the null hypothesis.

Key Concepts

Standardized Test Statistic (t)

A standardized test statistic, such as t, is a value derived from sample data that measures how far the sample mean is from the null hypothesis mean, expressed in terms of standard errors. It helps determine whether to reject the null hypothesis by comparing the calculated t value to critical values from the t-distribution.

Null Hypothesis (H0)

The null hypothesis (H0) is a statement that there is no effect or no difference, serving as a default position in hypothesis testing. It is tested against an alternative hypothesis (H1) and is typically rejected if the evidence from the data is strong enough, often determined by the significance level and the calculated test statistic.

Rejection Region

The rejection region is the area in the tails of the distribution where, if the test statistic falls within this area, the null hypothesis is rejected. In the context of a t-distribution, this region is determined by the significance level (alpha) and is critical for making decisions about the null hypothesis based on the calculated t value.