Question

Problem Scenario: Following is a problem description. For all hypothesis tests, you MUST state the statistical test you are using and use the P-VALUE METHOD through Microsoft Excel to make your decision. Show all steps, calculations, and work. For confidence intervals there is a specific Excel tool for each interval.  Treat each part of the question as a separate problem -- we use the same data set but are answering different “research questions”.

Many parts of cars are mechanically tested to be certain that they do not fail prematurely. In an experiment to determine which one of two types of metal alloy produces superior door hinges, 40 of each type were tested until they failed. To evaluate how long hinges made with the different alloys would last, the number of openings and closings was observed and recorded (to the closest 0.1 million). Car manufacturers consider any hinge that does not survive 1 million openings and closings to be a failure., A statistician has determined that the number of openings and closings is normally distributed.

NOTE: use ONLY the P-value method for hypothesis tests.

Number of Openings and Closings

Alloy 1				Alloy 2
1.5	1.5	0.9	1.3	1.4	0.9	1.3	0.8
1.8	1.6	1.3	1.5	1.3	1.3	0.9	1.4
1.6	1.2	1.2	1.8	0.7	1.2	1.1	0.9
1.3	0.9	1.5	1.6	1.2	0.8	1.2	1.1
1.2	1.3	1.4	1.4	0.8	0.7	1.1	1.4
1.1	1.5	1.1	1.5	1.1	1.4	0.8	0.8
1.3	0.8	0.8	1.1	1.3	1.1	1.5	0.9
1.1	1.6	1.6	1.3	1.4	1.2	1.3	1.6
0.9	1.4	1.7	0.9	0.6	0.9	1.8	1.4
1.1	1.3	1.9	1.3	1.5	0.8	1.6	1.3

Answer the following four parts of the question:

a) Can we conclude at the 5% significance level that the mean number of door openings and closings with hinges made from Alloy 1 is greater than 1.25 million?

b.) Can we conclude at the 10% significance level that the variance of the number of openings and closings with the hinges made from Alloy 2 is less than 0.035.

c.) Estimate with 90% confidence the difference in the number of openings and closings between hinges made with Alloy1 and hinges made with Alloy 2. Interpret the interval.

d.) The quality control manager is not only concerned about the openings and closings of the hinges but is also concerned about the proportion of hinges that fail. Can we infer at the 10% significance level that the proportion of hinges made with Alloy 2 that fail exceeds 18%?

Answer 1

a)

Using Excel

data -> data analysis -> t-Test: Two-Sample Assuming Equal Variances

t-Test: Two-Sample Assuming Equal Variances
	alloy 1	alloy 2
Mean	1.3275	1.145
Variance	0.078967949	0.086128205
Observations	40	40
Pooled Variance	0.082548077
Hypothesized Mean Difference	1.25
df	78
t Stat	-16.6161
P(T<=t) one-tail	0.0000
t Critical one-tail	1.6646
P(T<=t) two-tail	0.0000
t Critical two-tail	1.9908

TS = -16.6161

p-value = P( t > TS) = 1 {this is right-tailed test}

p-value > alpha (0.05)

hence we fail to reject the null hypothesis

b)

TS = (n-1)S^2/sigma^2

= 39*0.086128205/ 0.035

= 95.9714

p-value = P(chi^2 < 95.9714)

= =CHISQ.DIST(95.9714,39,1)

=0.9999

p-value > alpha (0.05)

hence we fail to reject the null hypothesis