The test statistic of z=-1.94 is obtained when testing the...

The test statistic of $z=-1.94$ is obtained when testing the claim that $p=3 / 8$. a. Identify the hypothesis test as being two-tailed, left-tailed, or right-tailed. b. Find the $P$ -value. (See Figure $8-3$ on page 364.) c. Using a significance level of $\alpha=0.05$, should we reject $H_{0}$ or should we fail to reject $H_{0}$ ?

The test statistic of $z=-1.94$ is obtained when testing the claim that $p=3 / 8$.
a. Identify the hypothesis test as being two-tailed, left-tailed, or right-tailed.
b. Find the $P$ -value. (See Figure $8-3$ on page 364.)
c. Using a significance level of $\alpha=0.05$, should we reject $H_{0}$ or should we fail to reject $H_{0}$ ?

Key Concepts

Decision Rule

The decision rule in hypothesis testing compares the P-value to the significance level. If the P-value is less than ?, the null hypothesis is rejected, indicating statistically significant evidence against it. Conversely, if the P-value is greater than or equal to ?, we fail to reject the null hypothesis, meaning there is insufficient evidence to support the alternative hypothesis.

Significance Level (?)

The significance level, denoted by ?, is the threshold probability set by the researcher to decide whether to reject the null hypothesis. It represents the risk of committing a Type I error—rejecting the null hypothesis when it is actually true. A common value for ? is 0.05, meaning there is a 5% risk of a Type I error.

P-value

The P-value is the probability, computed under the assumption that the null hypothesis is true, of obtaining a test statistic as extreme as or more extreme than the one observed. It quantifies the evidence against the null hypothesis. In two-tailed tests, the P-value is typically doubled to account for extreme values on both sides of the sampling distribution.

One?Tailed versus Two?Tailed Tests

The type of test – one-tailed or two-tailed – depends on the alternative hypothesis. In a one-tailed test, the alternative hypothesis specifies a direction of the effect (for example, less than or greater than a certain value), whereas a two-tailed test allows for deviation in either direction. Identifying the test type is vital because it determines how the rejection regions and P-value are calculated.

Hypothesis Testing

Hypothesis testing is the process of making decisions using data. It involves setting up a null hypothesis (often representing a statement of no effect or the status quo) and an alternative hypothesis (representing the effect or claim to be tested). The process then uses sample data and a test statistic to decide whether there is sufficient evidence to reject the null hypothesis in favor of the alternative.

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