To find the p-value for this two-tailed chi-squared test, we need to:

1. Identify the degrees of freedom ($df$) for the chi-squared distribution.
2. Use the given test statistic value to find the p-value.

Step 1: Calculate the degrees of freedom

The degrees of freedom for a chi-squared test are given by: $df = n - 1$ where $n$ is the sample size. Given $n = 88$, the degrees of freedom are: $df = 88 - 1 = 87$

Step 2: Find the p-value using the chi-squared distribution

For a two-tailed test, we need to find the probability that the chi-squared statistic is as extreme or more extreme than 12.126 in either tail.

Using a chi-squared distribution table or software with $df = 87$, we can find the p-value.

Calculating using statistical software or a calculator:

• The left-tail probability $P(\chi^2 < 12.126)$ can be obtained.
• The p-value for a two-tailed test is then twice the left-tail probability.

The p-value for the given test statistic (rounded to four decimal places) is approximately:

$P \approx 0.0006$

Therefore, the p-value for this hypothesis test is 0.0006.

Would you like more details on these calculations? Here are five related questions to help deepen your understanding:

1. How do you interpret p-values in hypothesis testing?
2. Why is it important to use the degrees of freedom when finding the p-value?
3. What is the significance of a two-tailed test compared to a one-tailed test?
4. How does sample size affect the degrees of freedom and p-value calculation?
5. Can you use chi-squared tests for non-normally distributed data?

Tip: When conducting a chi-squared test, ensure your sample size is sufficiently large, as small sample sizes can affect the accuracy of the test.