Question

8. (15 pts) The minimal Brinell hardness for a specific grade of ductile iron is 130. An engineer has a sample of 25 pieces of this type of iron that was subcritically annealed with the Brinell hardness values as given below. The engineer would like to know whether this annealing process results in the proper Brinell hardness on average.

   135 149 132 142 124 130 122 128 120 128 127 123 136 141 130 139 134 135 130 141 149 137 137 140 148

   State the appropriate hypotheses; conduct a hypothesis test using α = 0.05 utilizing the classical approach, confidence interval approach, or p-value approach; state the decision regarding the hypotheses; and make a conclusion.

Answer 1

The table given below ,

	X
	135	0.5184
	149	216.6784
	132	5.1984
	142	59.5984
	124	105.6784
	130	18.3184
	122	150.7984
	128	39.4384
	120	203.9184
	128	39.4384
	127	52.9984
	123	127.2384
	136	2.9584
	141	45.1584
	130	18.3184
	139	22.2784
	134	0.0784
	135	0.5184
	130	18.3184
	141	45.1584
	149	216.6784
	137	7.3984
	137	7.3984
	140	32.7184
	148	188.2384
Sum	3357	1625.04

From table ,

The sample mean is ,

The sample standard deviation is ,

Hypothesis: VS  

The critical value is ,

; From excel "=TINV(0.05,24)"

The test statistic is ,

The p-value is ,

p-value= ; From excel "=TDIST(2.6007,24,2)"

The 95% confidence interval is ,

Decision : 1) From critical value : Here , the value of the test statistic is lies in the rejection region.

Therefore , reject the null hypothesis.

2) From p-value : Here , p-value = 0.0157 <

Therefore , reject the null hypothesis.

3) From confidence interval : Here , the value of the does not lies in the 95% confidence interval.

Therefore , reject the null hypothesis.

Conclusion : Hence , There is not sufficient evidence to support the engineers claim that the  annealing process results in the proper Brinell hardness on average.