MHB Chloe's question via email about a p-value

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The discussion centers on calculating the p-value for a hypothesis test with null hypothesis H0: μ = 13 and alternative hypothesis Ha: μ < 13, using provided data. The test statistic is calculated as approximately -0.253, leading to an initial p-value estimate of about 0.401. A more precise p-value obtained through technology is approximately 0.399948, confirming the accuracy of the initial approximation. The conversation highlights the importance of using both manual calculations and technology for statistical analysis. Overall, the calculations demonstrate a solid understanding of hypothesis testing and p-value determination.
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I'm assuming the hypothesis test is

$\displaystyle H_0 : \mu = 13 \quad \quad H_a : \mu < 13 $

We are given $\displaystyle \mu = 13, \quad \sigma = 2.47, \quad \bar{x} = 12.86 , \quad n = 20 $.

The test statistic is

$\displaystyle \begin{align*} z &= \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}} \\
&= \frac{12.86 - 13}{\frac{2.47}{\sqrt{20}}} \\
&\approx -0.253\,481 \end{align*} $

Thus the p value is

$\displaystyle \begin{align*} p &= \textrm{Pr}\left( Z < -0.253\,481 \right) \\
&\approx \Phi \left( -0.25 \right) \textrm{ from the Z distribution table}\\
&= 0.401\,29 \end{align*} $

However, a CAS (or Linear Interpolation) could be used to get a more accurate value. Using technology, I find the p value to be $\displaystyle 0.399\,948 $, which our approximation is very close to.
 
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