To find the p-value for a two-tailed test using the z-score, you typically use the cumulative distribution function (CDF) of the standard normal distribution. Here’s the equation and steps to find the p-value:

Equation for the p-value

The p-value for a two-tailed test is given by: $\text{p-value} = 2 \times P(Z < -|z|)$

Where $|z|$ is the absolute value of the test statistic.

Steps to Find the p-value

1. Calculate the test statistic (z-score): $z = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{\hat{p}(1 - \hat{p}) \left( \frac{1}{n_1} + \frac{1}{n_2} \right)}}$

2. Find the cumulative probability associated with the z-score:

   • Use the standard normal distribution table, or
   • Use a statistical software or calculator to find $P(Z < z)$.

3. Calculate the p-value: $\text{p-value} = 2 \times P(Z < -|z|)$

Given our previous z-score of $-5.93$, let's find the p-value using this method.

Calculation of the p-value

1. Compute the test statistic (as previously calculated): $z = -5.93$

2. Find $P(Z < -5.93)$:

   • For a z-score of $-5.93$, the cumulative probability $P(Z < -5.93)$ is extremely small, essentially close to 0.

3. Calculate the two-tailed p-value: $\text{p-value} = 2 \times P(Z < -5.93)$ $\text{p-value} \approx 2 \times 0$ $\text{p-value} \approx 0$

Since the cumulative probability $P(Z < -5.93)$ is extremely small (close to 0), the p-value will also be extremely small, confirming our earlier conclusion that the p-value is effectively 0.

Using Statistical Software or Calculators

In practice, you can use statistical software or online calculators to find the exact p-value. For example:

• In R:

  RCopy code
  p_value <- 2 * pnorm(-abs(-5.93))

• In Python (using scipy):

  pythonCopy code
  from scipy.stats import norm
  p_value = 2 * norm.cdf(-abs(-5.93))

Summary

• Equation for p-value: $\text{p-value} = 2 \times P(Z < -|z|)$
• Test Statistic (z): $-5.93$
• p-value: Close to 0

Would you like further details or have any questions about this process? Here are some related questions you might consider:

1. How can I find the p-value using an online calculator?
2. What is the significance level, and how does it relate to the p-value?
3. How do I interpret the results in the context of my study?
4. What are type I and type II errors in hypothesis testing?
5. How can I perform hypothesis testing in software like R or Python?

Tip: When performing hypothesis testing, ensure that you correctly interpret the p-value in the context of your chosen significance level (e.g., 0.05) to make informed decisions about the null hypothesis.