To determine whether the null hypothesis will be rejected, we need to compare the given $X^2$ statistic (6.25) with the critical value for a chi-square distribution with 3 degrees of freedom at a significance level ($\alpha$) of 5%.

Step 1: Identify the critical value

From a chi-square table, the critical value for df = 3 and $\alpha = 0.05$ (right-tailed test) is approximately:

$X^2_{0.05, 3} = 7.815$

Step 2: Compare the test statistic to the critical value

• The given test statistic is 6.25.
• The critical value is 7.815.
• Since 6.25 < 7.815, we fail to reject the null hypothesis at the 5% significance level.

Step 3: Interpret the shaded area

The problem states that the shaded area (the p-value) is 0.1. Since 0.1 > 0.05, this confirms that we do not have enough evidence to reject the null hypothesis.

Conclusion:

The statement "The null hypothesis for a significance level alpha of 5% will be rejected." is False.

---

Would you like a detailed breakdown or a chi-square table reference?

Here are some related questions:

1. What is the chi-square distribution used for in hypothesis testing?
2. How do you calculate the p-value in a chi-square test?
3. What happens if the test statistic exceeds the critical value?
4. How does increasing the degrees of freedom affect the chi-square distribution?
5. What are the assumptions of a chi-square test?

Tip: Always compare the test statistic to the critical value when deciding whether to reject the null hypothesis.