A consumer’s organization wants to prove that the probability of a mobile phone

Question

A consumer’s organization wants to prove that the probability of a mobile phone from Company A breaking down in the first 24 months is more than 0.1 (i. e. ten percent).

1. Formulate the null and the alternative hypotheses. The parameter is the probability pi of the phone breaking down.

2. Explain in a few words why for this question, a lower bound but not an upper bound for pi is interesting for the consumer’s organization.

3. For this aim, 400 phones were tested with 45 being defective. Test the hypothesis with a one-sided binomial test in R (provide the R code). Report the test decision, a one-sided p-value and a lower confidence bound for pi at level 95%.

4. Now the organization also tests 400 phones from Company B in the same manner, observing 28 defective phones. First, compute the sample odds of the phone breaking down for each company and then compute and interpret the sample odds ratio.

Please note: “interpret” only means to give a sentence of the type: The odds of ... are ... times higher for Company ... than for Company ...

Explanation / Answer

Answer 1) H0 : (Pi) Probability of mobile phone from company A beaking down in the first 24 months is more than 0.1.

verses

Ha :  (Pi) Probability of mobile phone from company A beaking down in the first 24 months is not more than 0.1.

Here,H0 is null hypothesis and Ha is alternative hypothesis.

Answer 2) Here,lower bound but not an upper bound for pi is interesting for the consumer’s organization. because we consider a breaking down of mobile phone in company A so we are going to consider minimum time in which mobile will break down and we also have to test the same thing so that here lower bound is important and upper bound is not that much important.

answer 3) Binomail test

R commond-

> binom.test(355, 400, 0.1, conf.level=0.9,alternative="less")

Test result-

data: 355 and 400
number of successes = 355, number of trials = 400, p-value = 1
alternative hypothesis: true probability of success is less than 0.1
90 percent confidence interval:
0.0000000 0.9075116
sample estimates:
probability of success
0.8875

here p value is 1 and lower confidence bond is 0.

Answer 4) odds ratio=(a/b)/(c/d)

here, a=400, b=45,c=400 and d=28

so odds ratio=(400/45)/(400/28)

that is odds ratio=(400*28)/(45*400)

odds ratio=0.622

interpretation:-odds ratio is less than 1 mean there is a associated with lower odds. means there is very low association between company A and company B.

Answer 5) 95% confidence interval for the true odds ratio

confidence interval = log(0.622) +/- 1.96*sqrt(1/400+1/45+1/400+1/28 = (0.2062,0.2854).

exp(0.2062), exp(0.2854) =(0.38,1.02)

Interpretation:-In this study, odds for Probability of mobile phone from company A beaking down in the first 24 months is more than 0.1 compared to company B with 95% confidence the true odds ratio lies in the range of 0.38-1.02.

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