A machine puts out 16 imperfect articles in a sample of 500. After the machine is overhauled it puts 3 imperfect articles in a batch of 100. Has the machine been improved? Use 5% level of significance. Use p-value fot your conclusion and also find 88% confidence interval for difference between population proportion.

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Asked Dec 15, 2019
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A machine puts out 16 imperfect articles in a sample of 500. After the machine is overhauled it puts 3 imperfect articles in a batch of 100. Has the machine been improved? Use 5% level of significance. Use p-value fot your conclusion and also find 88% confidence interval for difference between population proportion.

 

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Expert Answer

Step 1

From the given information, the claim of the problem is the machine has been improved.

 

Null Hypothesis:

 

H0:p1=p2

 

Alternative Hypothesis:

 

Ha:p1-p2<0

 

Test Statistic:

 

From the given information, n1=500, x1=16, n2=100 and x2=3.

 

The pooled sample proportion is,

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X, +x, п+п, 16+3 500+100 = 0.032 п, 16 500 = 0.032 X2 п, 3 100 = 0.03 || ||

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Step 2

The test statistic is 0.104 and it is calculated below:

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Ztest a- (1– P) п п, 0.032 – 0.03 (0.032 (1-0.032) 500 100 0.002 0.0193 = 0.104 ||

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Step 3

p-value:

 

The p-value is 0.5414 and it is calculated below:

...
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p-value = P(z < Ziest ) [Since, the hypothesis is left tail test] = P(z <0.104) [=NORM.DIST(0.104,0,1,TRUE)] =0.5414

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