State your decision. Since the $$ p $$-value is greater than .01 , reject $$ H_{0} $$. Since the $$ p $$-value is greater than .01 , fail to reject $$ H_{0} $$. Since the $$ p $$-value is less than .01, fail to reject $$ H_{0} $$. Since the $$ p $$-value is less than .01 , reject $$ H_{0} $$.

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To determine whether to reject or fail to reject the null hypothesis ($$ H_0 $$) based on the $$ p $$-value and a specified significance level ($$ \alpha $$), follow these general rules: 1. **Compare the $$ p $$-value to $$ \alpha $$:** - **If $$ p $$-value $$\leq \alpha$$:** Reject $$ H_0 $$. - **If $$ p $$-value $$> \alpha$$:** Fail to reject $$ H_0 $$. Given your significance level $$ \alpha = 0.01 $$ (i.e., 1%), apply these rules to each scenario: 1. **Since the $$ p $$-value is greater than .01, reject $$ H_{0} $$.** - **Incorrect.** If $$ p $$-value $$> 0.01$$, you **fail to reject** $$ H_0 $$. 2. **Since the $$ p $$-value is greater than .01, fail to reject $$ H_{0} $$.** - **Correct.** When $$ p $$-value $$> 0.01$$, you **do not have enough evidence** to reject $$ H_0 $$. 3. **Since the $$ p $$-value is less than .01, fail to reject $$ H_{0} $$.** - **Incorrect.** If $$ p $$-value $$< 0.01$$, you **reject** $$ H_0 $$. 4. **Since the $$ p $$-value is less than .01, reject $$ H_{0} $$.** - **Correct.** When $$ p $$-value $$< 0.01$$, there is sufficient evidence to **reject** $$ H_0 $$. ### **Summary of Decisions:** - **If $$ p $$-value $$> 0.01$$:** **Fail to reject** $$ H_0 $$. - **If $$ p $$-value $$< 0.01$$:** **Reject** $$ H_0 $$. Based on your provided options: - **Option 2** and **Option 4** are **correct**. - **Option 1** and **Option 3** are **incorrect**. ### **Example:** Suppose you conducted a hypothesis test with $$ \alpha = 0.01 $$: - **Case 1:** $$ p $$-value = 0.05 - $$ 0.05 > 0.01 $$ ⇒ **Fail to reject** $$ H_0 $$. - **Case 2:** $$ p $$-value = 0.005 - $$ 0.005 < 0.01 $$ ⇒ **Reject** $$ H_0 $$. ### **Final Decision Statements:** - **If $$ p $$-value $$\leq$$ 0.01:** "Since the $$ p $$-value is less than .01, reject $$ H_{0} $$." - **If $$ p $$-value $$>$$ 0.01:** "Since the $$ p $$-value is greater than .01, fail to reject $$ H_{0} $$." Choose the statement that aligns with your specific $$ p $$-value and significance level. If the $$ p $$-value is greater than 0.01, fail to reject $$ H_0 $$. If the $$ p $$-value is less than 0.01, reject $$ H_0 $$.

State your decision. Since the \( p \)-value is greater than .01 , reject \( H_{0} \). Since the \( p \)-value is greater than .01 , fail to reject \( H_{0} \). Since the \( p \)-value is less than .01, fail to reject \( H_{0} \). Since the \( p \)-value is less than .01 , reject \( H_{0} \).

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