Question

According to an independent insurance agent, the following is a breakdown of the amount of life insurance purchased by males in the following age groups. The agent is interested in whether the age of the male and the amount of life insurance purchased are independent events. Conduct a test for independence at the 5% level.

Age of Males	None	$50,000 – $100,000	$100,001 – $150,000	$150,001 – $200,000	$200,000+
20 – 29	45	15	40	0	5
30 – 39	40	5	25	20	10
40 – 49	15	0	30	0	35
50+	40	30	10	10	10

• Part (a)

  State the null hypothesis.

  The amount of life insurance purchased is dependent on the age of the male.The amount of life insurance purchased is independent of the age of the male.    

• Part (b)

  State the alternative hypothesis.

  The amount of life insurance purchased is independent of the age of the male.The amount of life insurance purchased is dependent on the age of the male.    

• Part (c)

  What are the degrees of freedom?

• Part (d)

  State the distribution to use for the test.

  t₄

  t₁₂

      

• Part (e)

  What is the test statistic? (Round your answer to two decimal places.)

• Part (f)

  What is the p-value? (Round your answer to four decimal places.)
  
  
  Explain what the p-value means for this problem.If

  H₀

  is true, then there is a chance equal to the p-value that the value of the test statistic will be equal to or greater than the calculated value.If

  H₀

  is false, then there is a chance equal to the p-value that the value of the test statistic will be equal to or less than the calculated value.    If

  H₀

  is true, then there is a chance equal to the p-value that the value of the test statistic will be equal to or less than the calculated value.If

  H₀

  is false, then there is a chance equal to the p-value that the value of the test statistic will be equal to or greater than the calculated value.

• Part (g)

  Sketch a picture of this situation. Label and scale the horizontal axis, and shade the region(s) corresponding to the p-value. (Upload your file below.)

• Part (h)

  Indicate the correct decision ("reject" or "do not reject" the null hypothesis) and write the appropriate conclusion.(i) Alpha:
  α =
  
  (ii) Decision:

  reject the null hypothesisdo not reject the null hypothesis    

  
  (iii) Reason for decision:

  Since α < p-value, we do not reject the null hypothesis.Since α > p-value, we reject the null hypothesis.    Since α < p-value, we reject the null hypothesis.Since α > p-value, we do not reject the null hypothesis.

  
  (iv) Conclusion:

  There is sufficient evidence to conclude that the amount of life insurance purchased is dependent on the age of the male.There is not sufficient evidence to conclude that the amount of life insurance purchased is dependent on the age of the male.   

Answer 1

• Part (a)

  State the null hypothesis.

  The amount of life insurance purchased is independent of the age of the male.    

• Part (b)

  State the alternative hypothesis.

  The amount of life insurance purchased is dependent on the age of the male.    

• Part (c)

  What are the degrees of freedom? 12

• Part (d)

  State the distribution to use for the test.

  t₁₂

      

• Part (e)

  What is the test statistic? (Round your answer to two decimal places.) 151.98

• Part (f)

  What is the p-value? (Round your answer to four decimal places.)
  0.0000
  
  Explain what the p-value means for this problem.If

  H₀

  is true, then there is a chance equal to the p-value that the value of the test statistic will be equal to or greater than the calculated value.

• Part (g)

  Sketch a picture of this situation. Label and scale the horizontal axis, and shade the region(s) corresponding to the p-value. (Upload your file below.)

Distribution Plot Chi-Square, df=12 0.09 0.08 0.07 0.06 0.05 Density 0.04 0.03 0.02 0.01 2.24e-026 0.00 х

• Part (h)

  Indicate the correct decision ("reject" or "do not reject" the null hypothesis) and write the appropriate conclusion.(i) Alpha:
  α = 0.05
  
  (ii) Decision:

  reject the null hypothesis

  
  (iii) Reason for decision:

  Since α < p-value, we reject the null hypothesis.

  
  (iv) Conclusion:

  There is sufficient evidence to conclude that the amount of life insurance purchased is dependent on the age of the male.

• 		None	$50,000 – $100,000	$100,001 – $150,000	$150,001 – $200,000	$200,000+	Total
  20 – 29	Observed	45	15	40	0	5	105
  	Expected	38.18	13.64	28.64	8.18	16.36	105.00
  	O - E	6.82	1.36	11.36	-8.18	-11.36	0.00
  	(O - E)² / E	1.22	0.14	4.51	8.18	7.89	21.94
  30 – 39	Observed	40	5	25	20	10	100
  	Expected	36.36	12.99	27.27	7.79	15.58	100.00
  	O - E	3.64	-7.99	-2.27	12.21	-5.58	0.00
  	(O - E)² / E	0.36	4.91	0.19	19.13	2.00	26.59
  40 – 49	Observed	15	0	30	0	35	80
  	Expected	29.09	10.39	21.82	6.23	12.47	80.00
  	O - E	-14.09	-10.39	8.18	-6.23	22.53	0.00
  	(O - E)² / E	6.83	10.39	3.07	6.23	40.72	67.24
  50+	Observed	40	30	10	10	10	100
  	Expected	36.36	12.99	27.27	7.79	15.58	100.00
  	O - E	3.64	17.01	-17.27	2.21	-5.58	0.00
  	(O - E)² / E	0.36	22.29	10.94	0.63	2.00	36.22
  Total	Observed	140	50	105	30	60	385
  	Expected	140.00	50.00	105.00	30.00	60.00	385.00
  	O - E	0.00	0.00	0.00	0.00	0.00	0.00
  	(O - E)² / E	8.77	37.73	18.71	34.17	52.62	151.98
  		151.98	chi-square
  		12	df
  		2.24E-26	p-value