d) Will the researcher reject the null​ hypothesis?

RejectReject

Upper H 0H0

because

chi Subscript 0 Superscript 2χ20less than<chi Subscript 0.01 Superscript 2χ20.01.

Do not rejectDo not reject

Upper H 0H0

because

chi Subscript 0 Superscript 2χ20greater than>chi Subscript 0.01 Superscript 2χ20.01.

RejectReject

Upper H 0H0

because

chi Subscript 0 Superscript 2χ20greater than>chi Subscript 0.01 Superscript 2χ20.01.

Do not rejectDo not reject

Upper H 0H0

because

chi Subscript 0 Superscript 2χ20less than<chi Subscript 0.01 Superscript 2χ20.01.

In hypothesis testing using the chi-square (\( \chi^2 \)) test, the decision to **reject** or **not reject** the null hypothesis \( H_0 \) depends on the comparison between the test statistic (\( \chi^2_0 \)) and the critical value (\( \chi^2_{\alpha} \)) for a given significance level \( \alpha \).

- **Reject \( H_0 \)** if the test statistic \( \chi^2_0 \) is **greater than** the critical value \( \chi^2_{\alpha} \), meaning that the result is significant, and the null hypothesis is unlikely to be true.
- **Do not reject \( H_0 \)** if the test statistic \( \chi^2_0 \) is **less than** the critical value \( \chi^2_{\alpha} \), meaning that there is insufficient evidence to reject the null hypothesis.

Let's match this logic to your answer options:

1. **Reject \( H_0 \)** because \( \chi^2_0 < \chi^2_{0.01} \): This is incorrect, because when the test statistic is less than the critical value, we should **not reject** the null hypothesis.
2. **Do not reject \( H_0 \)** because \( \chi^2_0 > \chi^2_{0.01} \): This is also incorrect. If the test statistic is greater than the critical value, we should **reject** the null hypothesis.
3. **Reject \( H_0 \)** because \( \chi^2_0 > \chi^2_{0.01} \): This is correct, as when the test statistic exceeds the critical value, we reject the null hypothesis.
4. **Do not reject \( H_0 \)** because \( \chi^2_0 < \chi^2_{0.01} \): This is correct, as when the test statistic is less than the critical value, we do not reject the null hypothesis.

So, the correct conclusion depends on whether \( \chi^2_0 \) is greater than or less than \( \chi^2_{0.01} \). If \( \chi^2_0 > \chi^2_{0.01} \), we **reject** \( H_0 \); otherwise, we **do not reject** \( H_0 \).

Would you like more details or have any questions?

Here are 5 related questions for further exploration:
1. How is the chi-square critical value \( \chi^2_{\alpha} \) determined?
2. What are the assumptions required to perform a chi-square test?
3. How does the sample size affect the chi-square test statistic?
4. Can the chi-square test be used for continuous data?
5. What happens if the expected frequency in a chi-square test is too small?

**Tip:** Always verify the degrees of freedom before using the chi-square distribution table to find the critical value.

Will the researcher reject the null hypothesis? Reject H0 because χ²0 < χ²0.01. Do not reject H0 because χ²0 > χ²0.01. Reject H0 because χ²0 > χ²0.01. Do not reject H0 because χ²0 < χ²0.01.

Understand the decision-making process in a chi-square test for rejecting or not rejecting the null hypothesis. This guide explains how to compare the test statistic (χ²0) with the critical value (χ²0.01) at a 1% significance level and includes step-by-step solutions. Ideal for undergraduate statistics students.

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