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


c. t = -2.096

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 value(s) and the significance level (α).
2. Step 2: Identify the given information. The test statistic t₀ is -2.086, as shown in the graph. The graph also displays the t-distribution curve, with the shaded region representing the rejection area for the null hypothesis.
3. Step 3: Determine the critical value(s). The critical value(s) depend on the significance level (α) and the degrees of freedom (df). These values are typically obtained from a t-distribution table or software. Compare t₀ to the critical value(s).
4. Step 4: Analyze the graph. The shaded region to the left of the critical value indicates the rejection region for the null hypothesis. If t₀ falls within this region, the null hypothesis is rejected.
5. Step 5: Conclude based on the comparison. If t₀ = -2.086 is less than the critical value (e.g., -t_critical), then the null hypothesis is rejected. Otherwise, it is not rejected. Ensure you verify the significance level and degrees of freedom to confirm the critical value.

Key Concepts

Standardized Test Statistic (t)

A standardized test statistic, such as t, is used in hypothesis testing to determine how far a sample mean is from the population mean in terms of standard errors. It is calculated by taking the difference between the sample mean and the population mean, divided by the standard error of the mean. The value of t helps assess whether to reject the null hypothesis based on its position relative to critical values.

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 H0 is based on the calculated test statistic and its comparison to critical values from the relevant statistical distribution.

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. In a t-distribution, this region is determined by the significance level (alpha) and the degrees of freedom. For the given t value of -2.096, if it falls within the rejection region (to the left of -2.086 in this case), it indicates sufficient evidence to reject the null hypothesis.