From a random sample of 673 items made by a particular manufacturing process, it

Question

From a random sample of 673 items made by a particular manufacturing process, it is found that 27 are defective. (a) (1 mark) Find a 99.5% confidence interval for the proportion of defective items made by the process. (Also include the commands and output.) (b) (1 mark) Give the command and output to test the alternative hypothesis that the proportion of defective items made by the process is greater than than 0.03. (c) (1 mark) What is the p-value for our test? (d) (1 mark) What is the strength of evidence we have found against Ho?

Explanation / Answer

Here, we have to use the R-commands for finding the confidence interval and test statistic with p-value.

Part a

The 99.5% confidence interval for the population proportion by using R-codes is given as below:

> c=0.995

> alpha = 1 - c

> alpha

[1] 0.005

> alpha/2

[1] 0.0025

> 1-alpha/2

[1] 0.9975

> qnorm(0.9975)

[1] 2.807034

> x=27

> n=673

> pbar=x/n

> pbar

[1] 0.04011887

> SE=sqrt(pbar*(1-pbar)/n)

> SE

[1] 0.007564418

> E=qnorm(0.9975)*SE

> E

[1] 0.02123358

> CI = pbar + c(-E,E)

> CI

[1] 0.01888529 0.06135245

Confidence interval = (0.01888529, 0.06135245)

Part b

Here, we have to test the alternative hypothesis Ha: p>0.03

R commands are given as below:

> x=27

> n=673

> pbar=x/n

> p=0.03

> SE=sqrt(p*(1-p)/n)

> Z=(pbar-p)/SE

> PV=dnorm(Z)

> Z

[1] 1.538839

> PV

[1] 0.1220956

Do not reject H0

Part c

P-value = 0.1220956

Part d

Strength of evidence is given as below:

Strength of evidence = 1 - 0.1220956 = 0.8779044 or 87.79%

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