Question

Calculate the p-value for the following conditions and determine whether or not to reject the null hypothesis. a) one-tail (upper) test, zp-2.51, and α-0.10 b) one-tail (lower) test, zp--1 .34, and α: 0.05 c) two-tail test, Zp-_ 1.89, and α: 0.05 d) two-tail test, z1.58, and a 0.02 Click here to view the first page of the standard normal table, Click here to view the second page of the standard normal table. a) p-value (Round to four decimal places as needed.) Determine whether or not to reject the null hypothesis. Choose the correct answer below. OA. Reject Ho, since the p-value is greater than or equal to the significance level a. B. Do not reject Ho, since the p-value is greater than or equal to the significance level α. ° C. Reject Ho, since the p-value is less than the significance level α. D. Do not reject Ho, since the p-value is less than the significance level α. b) p-value(Round to four decimal places as needed.) Determine whether or not to reject the null hypothesis. Choose the correct answer below. O A. Reject Ho, since the p-value is greater than or equal to the significance level α. O B. Reject Ho, since the p-value is less than the significance level α. C. Do not reject Ho, since the p-value is less than the significance level α. 0 D. Do not reject Ho, since the p-value is greater than or equal to the significance level α.

Answer 1

Solution :

We have to calculate the p-values for each of the following conditions and determine whether or not to Reject the Null Hypothesis.

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(a) To find the p - value for the One - Tail (upper) Test, Z_p = 2.51 and $\alpha$ = 0.10 :

We have the following information from the given data !

$Test=One-Tail\ (upper)\ \ ;\ \ Z_p=2.51\ \ ;\ \ \alpha=0.10$

$Thus,\ p-value=P(Z_{\alpha}>Z_p)=1-P(Z_\alpha\leq Z_p)$

$=1-P(Z_\alpha\leq 2.51)$

$=1-\Phi(2.51)\ \ \ [where,\ \Phi(*)=Dist\overline{n}\ Func.\ for\ N(0,1)]$

  

   $=0.0060366$

$\mathbf{Thus,\ p-value=0.0060...................(Ans)}$

$Now,\ clearly,\ 0.0060<0.10\ \Rightarrow \ p-value<\alpha\ \Rightarrow Reject\ H_0$

Conclusion : Thus , the Correct Conclusion for the above situtation is ,

Option (C) Reject H₀ , since the p-value is less than the significance level $\alpha$ ............... (Ans)

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(b) To find the p - value for the One - Tail (lower) Test, Z_p = -1.34 and $\alpha$ = 0.05 :

We have the following information from the given data !

$Test=One-Tail\ (lower)\ \ ;\ \ Z_p=-1.34\ \ ;\ \ \alpha=0.05$

$Thus,\ p-value=P(Z_{\alpha}\leq Z_p)$

$=P(Z_\alpha\leq -1.34)$

$=\Phi(-1.34)\ \ \ [where,\ \Phi(*)=Dist\overline{n}\ Func.\ for\ N(0,1)]$

  

$\mathbf{Thus,\ p-value=0.0901...................(Ans)}$

$Now,\ clearly,\ 0.0901>0.05\ \Rightarrow \ p-value>\alpha\ \Rightarrow Do\ not\ Reject\ H_0$

Conclusion : Thus , the Correct Conclusion for the above situtation is ,

Option (D) Do not Reject H₀ , since the p-value is greater than or equal to the significance level $\alpha$ ............... (Ans)

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(c) To find the p - value for the Two - Tail Test, Z_p = -1.89 and $\alpha$ = 0.05 :

We have the following information from the given data !

$Test=Two-Tail\ \ \ ;\ \ Z_p=-1.34\ \ ;\ \ \alpha=0.05$

$Thus,\ p-value=2*P(Z_{\alpha}>|\ Z_p\ |)$

$=2*P(Z_\alpha> |-1.89 |)$

$=2*\Phi(-|-1.89|)\ \ \ [\Phi(*)=Dist\overline{n}\ Func.\ for\ N(0,1)]$

$\mathbf{Thus,\ p-value=0.0588...................(Ans)}$

$Now,\ clearly,\ 0.0588>0.05\ \Rightarrow \ p-value>\alpha\ \Rightarrow Do\ not\ Reject\ H_0$

Conclusion : Thus , the Correct Conclusion for the above situtation is ,

Option (B) Do not Reject H₀ , since the p-value is greater than or equal to the significance level $\alpha$ ............... (Ans)

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(d) To find the p - value for the Two - Tail Test, Z_p = 1.58 and $\alpha$ = 0.02 :

We have the following information from the given data !

$Test=Two-Tail\ \ \ ;\ \ Z_p=1.58\ \ ;\ \ \alpha=0.02$

$Thus,\ p-value=2*P(Z_{\alpha}>|\ Z_p\ |)$

$=2*P(Z_\alpha> |1.58|)$

$=2*\Phi(-|1.58|)\ \ \ [ \Phi(*)=Dist\overline{n}\ Func.\ for\ N(0,1)]$

$\mathbf{Thus,\ p-value=0.1141...................(Ans)}$

$Now,\ clearly,\ 0.1141>0.02\ \Rightarrow \ p-value>\alpha\ \Rightarrow Do\ not\ Reject\ H_0$

Conclusion : Thus , the Correct Conclusion for the above situtation is ,

Option (D) Do not Reject H₀ , since the p-value is greater than or equal to the significance level $\alpha$ ............... (Ans)

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