Question

for each of the following, determine if the observed p - value would result in rejecting the null hypothesis, ( h_0 ), at the specified significance level, ( alpha ).
a. ( p - value = 0.087, alpha = 0.10 ): ( h_0 )
b. ( p - value = 0.087, alpha = 0.05 ): ( h_0 )
c. ( p - value = 0.042, alpha = 0.01 ): ( h_0 )
d. ( p - value = 0.0495, alpha = 0.05 ): ( h_0 )
e. ( p - value = 0.006, alpha = 0.001 ): ( h_0 )
f. ( p - value = 0.142, alpha = 0.10 ): ( h_0 )
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Explanation:

Step1: Recall the decision rule

If \(P - value\leq\alpha\), reject \(H_0\). If \(P - value>\alpha\), do not reject \(H_0\).

Step2: Apply the rule to each case

• Case a: \(P - value = 0.087\), \(\alpha=0.10\). Since \(0.087<0.10\), reject \(H_0\).
• Case b: \(P - value = 0.087\), \(\alpha = 0.05\). Since \(0.087>0.05\), do not reject \(H_0\).
• Case c: \(P - value = 0.042\), \(\alpha=0.01\). Since \(0.042>0.01\), do not reject \(H_0\).
• Case d: \(P - value = 0.0495\), \(\alpha=0.05\). Since \(0.0495<0.05\), reject \(H_0\).
• Case e: \(P - value = 0.006\), \(\alpha=0.001\). Since \(0.006>0.001\), do not reject \(H_0\).
• Case f: \(P - value = 0.142\), \(\alpha=0.10\). Since \(0.142>0.10\), do not reject \(H_0\).

Answer:

a. Reject \(H_0\)
b. Do not reject \(H_0\)
c. Do not reject \(H_0\)
d. Reject \(H_0\)
e. Do not reject \(H_0\)
f. Do not reject \(H_0\)