The alternative hypothesis is [tex]H_{1}: p = 2\% - 157[/tex], and the test statistic is [tex]2\% - 157[/tex]. Find the p-value of the large sample one proportion z-test.

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11-04-2025
Mathematics

High School

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The alternative hypothesis is [tex]H_{1}: p = 2\% - 157[/tex], and the test statistic is [tex]2\% - 157[/tex]. Find the p-value of the large sample one proportion z-test.

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muhammadaneesamjad muhammadaneesamjad · Answer 1 · 2024-01-10T15:06:54-05:00

The p-value of the large sample one proportion z-test is approximately 0.022.

In hypothesis testing, the p-value represents the probability of observing a test statistic as extreme as, or more extreme than, the one obtained from the sample data, under the assumption that the null hypothesis is true. In this scenario, the alternative hypothesis suggests that the population proportion is different from the null hypothesis value. Given a test statistic of 2% (or 0.02), to find the p-value, we look up the corresponding probability from the standard normal distribution table or calculate it using statistical software.

Since the alternative hypothesis is two-sided, we consider the area in both tails of the standard normal distribution curve. For a test statistic of 2% (or 0.02), the corresponding z-score can be calculated. Then, using the standard normal distribution table or a calculator, we find the area under the curve beyond the absolute value of this z-score. This area represents the probability of observing a test statistic as extreme as 2% or more extreme, assuming the null hypothesis is true. Therefore, the obtained p-value of approximately 0.022 indicates that there is moderate evidence against the null hypothesis at a significance level of 0.05.