Determine the p -value that would be used to test the...

Determine the $p$ -value that would be used to test the following hypotheses when $z$ is used as the test statistic. a. $\quad H_{o}: p_{1}=p_{2}$ versus $H_{a}: p_{1}>p_{2},$ with $z *=2.47$ b. $\quad H_{o}: p_{A}=p_{B}$ versus, $H_{a}: p_{A} \neq p_{B},$ with $z \star=-1.33$ c. $\quad H_{a}: p_{1}-p_{2}=0$ versus $H_{a}: p_{1}-p_{2}<0,$ with $z *=-0.85$ d. $\quad H_{o}: p_{m}-p_{f}=0$ versus $H_{a}: p_{m}-p_{f}>0,$ with $z *=3.04$

Determine the $p$ -value that would be used to test the following hypotheses when $z$ is used as the test statistic.
a. $\quad H_{o}: p_{1}=p_{2}$ versus $H_{a}: p_{1}>p_{2},$ with $z *=2.47$
b. $\quad H_{o}: p_{A}=p_{B}$ versus, $H_{a}: p_{A} \neq p_{B},$ with $z \star=-1.33$
c. $\quad H_{a}: p_{1}-p_{2}=0$ versus $H_{a}: p_{1}-p_{2}<0,$ with $z *=-0.85$
d. $\quad H_{o}: p_{m}-p_{f}=0$ versus $H_{a}: p_{m}-p_{f}>0,$ with $z *=3.04$

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to assess whether there is sufficient evidence to reject a stated null hypothesis in favor of an alternative hypothesis. It involves setting up specific hypotheses, selecting an appropriate test statistic, and using the probability distribution under the null hypothesis to determine the likelihood of the observed outcome.

P-value

The p-value is the probability of obtaining a test statistic as extreme as, or more extreme than, what was observed, given that the null hypothesis is true. It quantifies the strength of the evidence against the null hypothesis; smaller p-values indicate stronger evidence and can lead to the rejection of the null hypothesis.

One-tailed and Two-tailed Tests

One-tailed tests assess deviations in a single direction (either above or below a specified value), while two-tailed tests evaluate deviations in both directions (both above and below). The choice between one-tailed and two-tailed testing affects the calculation of the p-value, as one-tailed tests allocate the entire significance level to one side of the distribution, whereas two-tailed tests split it across both tails.

Z-test and the Standard Normal Distribution

The z-test is a statistical procedure that uses the z-score, which represents the number of standard deviations an observed sample statistic is from the population mean, to compare a sample statistic against a hypothesized parameter. It relies on the standard normal distribution (mean 0 and standard deviation 1) to determine the probabilities or p-values associated with the observed test statistic.