Question

A company manufacturing biomedical devices is involved in a legal case on the basis that a number of the devices have failed, and needed replacement, after too short a lifetime in use. The company claims to produce devices with median lifetime 96 months. However, twelve devices have reported failure times (in months) of 80.45, 87.01, 92.58, 91.66, 88.42, 96.47, 80.22, 87.41, 94.54, 95.36, 80.96, 96.66. Using a suitable nonparametric test, assess the extent of the statistical evidence supporting the legal case. State precisely the test used, the null and alternative hypotheses, and explain clearly how you complete the test.

Answer 1

The test we are going to carry out is hypothesis testing for median for a single sample.

We can choose a sign test which is a type of nonparametric test for median.

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Hypothesis:

Null hypothesis: The median lifetime is 96 months

Alternate hypothesis: The median lifetime is not equal to 96 months.

H0: m = 96

Ha: m 96 (Not equal to means that this is a two-sided test)

To do a sign test, we count number of observations less than the hypothesized median value and number of observations greater than the hypothesized median value. Let us name them as N- and N+

N- = 10

N+ = 2 [There are only 2 values of 96.47 and 96.66 that are greater than 96]

Test statistic: S = Max(N-, N+) = 10 [Greater of the two N- and N+]

So, S = 10

Now, we have to find the p-value.

If our null hypothesis H0 is true, then

p-value = 2 P[X >= S]

where X follows a binomial distribution with n = 12 and p = 0.5 [n = Number of observations]

P[X >= 10] = 1 - P[X < 10] = 1 - P[X<=9] = 1 - 0.9807 = 0.0193

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[Now, I hope it is clear why P[X <10] = P[X <=9]; This is because these are discrete values and can take integer values only.

Also, P[X<=9] can be found out by using excel formula: =BINOM.DIST(9,12,0.5,TRUE)

We put true because we need cumulative results i.e. P[X = 0] + P[X = 1] + .... P[X = 9]. Putting false will just give P[X = 9]

Alternatively, you can do it by hand as well by using the binomial formula for P[X = 0] + P[X = 1] + ..... P[X = 9] by calculating each of these separately by using binomial formula with n= 12 and p=0.5 and adding them all up]

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p = 2*P[X >= 10] = 2*0.0193 = 0.0386

Hence, now we know that our p-value = 0.0386

This value is less than 0.05 {Choosing significance level, alpha = 0.05]

Hence, there is enough evidence to reject the null hypothesis and conclude that the median lifetime of the devices are not equal to 96 months as claimed by the company.

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Thanks. Please let me know if you need any additional clarifications.

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