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


b. t = 1.42

Verified step by step guidance

1. Step 1: Understand the context of the problem. The standardized test statistic t = 1.42 is given, and we need to determine whether it allows us to reject the null hypothesis. This decision depends on the critical value and the significance level (α) of the test.
2. Step 2: Analyze the graph provided. The graph shows a t-distribution with the test statistic t₀ = 1.402 marked. The shaded region represents the area in the tail of the distribution, which corresponds to the p-value or the critical region.
3. Step 3: Compare the test statistic t₀ = 1.402 to the critical value. If the test statistic falls within the critical region (shaded area), we reject the null hypothesis. Otherwise, we fail to reject it. The critical value depends on the degrees of freedom and the significance level (α).
4. Step 4: Determine the significance level (α) and the corresponding critical value. If the graph or problem specifies α (e.g., 0.05), use a t-table or statistical software to find the critical value for the given degrees of freedom.
5. Step 5: Conclude based on the comparison. If t₀ = 1.402 is less than the critical value, the null hypothesis cannot be rejected. If t₀ is greater than the critical value, the null hypothesis is rejected. Ensure the interpretation aligns with the graph and the problem context.

Key Concepts

Standardized Test Statistic (t)

A standardized test statistic, such as t, is used in hypothesis testing to determine how far a sample statistic is from the null hypothesis value, measured in standard errors. The t statistic is particularly useful when the sample size is small and the population standard deviation is unknown. It helps assess whether to reject the null hypothesis based on the calculated value and the critical values from the t-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. Researchers aim to gather evidence against H0 to support an alternative hypothesis (H1). The decision to reject or fail to reject H0 is based on the comparison of the test statistic to critical values, which define the rejection region.

Rejection Region

The rejection region is the area in the tails of the probability distribution where, if the test statistic falls, the null hypothesis is rejected. This region is determined by the significance level (alpha), which is the probability of making a Type I error. In the context of the t-distribution, the rejection region is defined by critical t-values, and understanding its location helps in making decisions about the null hypothesis.