A student performs an independence test using technology to see if pet ownership is independent of relationship status. They get the following results: ${\chi }^2=0.545$ & $p=0.7614$. What can they conclude about pet ownership and relationship status?

• A. Reject $H_0$ since there is enough evidence to suggest that pet ownership and relationship status are dependent.
• B. Fail to reject $H_0$ since there is not enough evidence to suggest that pet ownership and relationship status are dependent.
• C. Reject $H_0$ since there is not enough evidence to suggest that pet ownership and relationship status are dependent.
• D. Fail to reject $H_0$ since there is enough evidence to suggest that pet ownership and relationship status are dependent.

Answer: (B)

Verified step by step guidance

1. Identify the null hypothesis \( H_0 \) and the alternative hypothesis \( H_a \). For a test of independence, \( H_0 \) states that pet ownership and relationship status are independent, while \( H_a \) states that they are dependent.
2. Look at the given test statistic \( \chi^2 = 0.545 \) and the p-value \( p = 0.7614 \). The p-value represents the probability of observing a test statistic as extreme as, or more extreme than, the one calculated, assuming \( H_0 \) is true.
3. Compare the p-value to the significance level \( \alpha \) (commonly 0.05). If \( p \leq \alpha \), reject \( H_0 \); if \( p > \alpha \), fail to reject \( H_0 \).
4. Since the p-value \( 0.7614 \) is greater than the typical significance level \( 0.05 \), there is not enough evidence to reject the null hypothesis.
5. Conclude that we fail to reject \( H_0 \), meaning there is not enough evidence to suggest that pet ownership and relationship status are dependent; they appear to be independent based on this test.