For the hypothesis test H0: μ=5 against H1: μ<5 and variance known, calculate the P -value for e

step 1 Here we are given a hypothesis test for which the no hypothesis is that the mean is equal to 5,

For the hypothesis test H0: μ=5 against H1: μ<5 and variance...

For the hypothesis test $H_{0}: \mu=5$ against $H_{1}: \mu<5$ and variance known, calculate the $P$ -value for each of the following test statistics.
(a) $z_{0}=2.05$
(b) $z_{0}=-1.84$
(c) $z_{0}=0.4$

Key Concepts

Standard Normal Distribution

The standard normal distribution is a special case of the normal distribution with a mean of zero and a standard deviation of one. It is extensively used in hypothesis testing, particularly in Z-tests, to determine the probability associated with a test statistic by transforming data to a common scale.

Z-Test

A Z-test is a statistical test used to determine whether there is a significant difference between sample statistics and population parameters under the assumption that the underlying distribution is normal, particularly when the variance is known. The test statistic follows a standard normal distribution (mean zero and standard deviation one), making it simple to compute probabilities and critical values.

P-Value

The p-value is the probability of obtaining a test statistic at least as extreme as the one that was actually observed, under the assumption that the null hypothesis is true. In a one-tailed test, the p-value corresponds to the probability of observing a value in the specified tail, providing a measure of the strength of evidence against the null hypothesis.

One-Tailed Test

A one-tailed test is a type of hypothesis test in which the alternative hypothesis specifies that the parameter of interest is either greater than or less than a certain value, directing all analysis to only one side of the probability distribution. This is particularly important when the research question has a directional focus, such as testing if the mean is less than a specified value.

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence in a sample of data to support a particular claim about a population parameter. It involves formulating a null hypothesis (typically a statement of no effect or standard condition) and an alternative hypothesis (the statement one is trying to find evidence for) and then using test statistics to decide whether to reject the null hypothesis.

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