A hypothesis will be used to test that a population mean...

A hypothesis will be used to test that a population mean equals 5 against the alternative that the population mean is less than 5 with known variance $\sigma .$ What is the critical value for the test statistic $Z_{0}$ for the following significance levels?
(a) 0.01
(b) 0.05
(c) 0.10

Key Concepts

Significance Level

The significance level, often denoted by alpha, is the probability of rejecting the null hypothesis when it is actually true. It quantifies the tolerance for Type I error, and common significance levels include 0.01, 0.05, and 0.10, which indicate the strictness of the test.

Critical Value

The critical value is the threshold in hypothesis testing that defines the boundary of the region where the null hypothesis will be rejected. It corresponds to a specified significance level and represents the point beyond which the probability of obtaining a test statistic is less than the significance level.

Z-Test

A Z-test is a type of statistical test used when the population variance is known and the sample size is large or the population is normally distributed. It employs the standard normal distribution to calculate probabilities and determine the significance of the test statistic.

Test Statistic

A test statistic is a standardized value calculated from sample data during hypothesis testing. Under the assumption that the null hypothesis is true, the test statistic follows a known probability distribution, which is then used to assess the likelihood of observing the sample data.

One-Tailed Test

A one-tailed test is used when the alternative hypothesis specifies the direction of the effect (for example, that the population mean is less than or greater than a certain value). It focuses on one end of the probability distribution, which affects the placement of the critical region used to determine statistical significance.

Null and Alternative Hypotheses

The null hypothesis is the statement that there is no difference or that the parameter equals a specific value, while the alternative hypothesis contradicts this and suggests a different condition or effect. These hypotheses form the basis of testing and determining whether there is enough evidence to reject the null hypothesis in favor of the alternative.

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population parameter based on sample data. It involves setting up a null hypothesis, which is the assumption of no effect or a specific value, and an alternative hypothesis, which represents a different claim that is tested against the null hypothesis.

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