Determine whether you would reject or fail to reject the...

Determine whether you would reject or fail to reject the null hypothesis in the following situations:
a. $z=1.99$, two-tailed test at $\alpha=0.05$
b. $z=0.34, z *=1.645$
c. $p=0.03, \alpha=0.05$
d. $p=0.015, \alpha=0.01$

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions based on data. It involves formulating a null hypothesis (typically a statement of no effect or no difference) and an alternative hypothesis, then using sample data to determine whether there is enough evidence to reject the null hypothesis using a predefined significance level.

Significance Level (?)

The significance level, denoted by ?, is the probability threshold for rejecting the null hypothesis. It represents the maximum risk of committing a Type I error—the error of rejecting a true null hypothesis. Common levels include 0.05 or 0.01, which set the standard for how strong the evidence must be before rejecting H?.

p-value

The p-value is the probability of observing test results at least as extreme as the results actually observed, under the assumption that the null hypothesis is true. If the p-value is less than or equal to the significance level, there is sufficient evidence to reject the null hypothesis. Otherwise, one fails to reject H?.

Critical Value

A critical value is a point on the test statistic distribution that is compared to the calculated statistic to decide whether to reject the null hypothesis. Critical values correspond to the predefined significance level and can differ based on whether the test is one-tailed or two-tailed.

Two-Tailed Test

A two-tailed test is a method used when the alternative hypothesis is non-directional, meaning that it does not specify the direction of the effect. In this test, the rejection regions are on both ends of the distribution, and the critical values are determined accordingly by splitting the significance level between the two tails of the distribution.

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