Question

Please write 300-400 words

Suppose a manufacturer of the XJ35 battery claims the mean life of the battery is

500 days with a standard deviation of 25 days. You are the buyer of this battery

and you think this claim is inflated. You would like to test your belief because

without a good reason you can’t get out of your contract.

What do you do?

How do you do it?

Who would you tell?

Answer 1

Steps:

1. First of all, we specify our variable under study. Here it is x or 'life of the battery'. Therefore the assumed mean, or µ₀=500 and σ=25.
2. At this point, we need to collect a sample of battery lives large enough to test for significance of the difference between the actual and the assumed means. Hence, we collect a large sample, i.e. the lives in days of 30 or more batteries. For ease of calculation in this case, let n be equal to 30.
3. Here, H₀: µ=µ₀=500 and H₁: µ<µ₀=500
4. Say this sample gives a mean of 480. If this number was 150, 183 or 310, we could have rejected the null hypothesis right away. However, since 480 looks close enough to 500, we test this statistically using the z-statistic.
5. z= (µ-µ₀)/(σ/sqrt(n)) = (480-500)/(25/(30^0.5)) = -4.38 which is more than two standard deviations away from the mean.
6. Testing this in R, using pnorm(480,500,25/sqrt(30)) we get a value smaller than 0.01, which will be our p-value. This means that, when the population mean is 500, there is less than 1% probability that we would find a sample mean smaller than 480.
7. To see whether the above probability is small enough, we check at a particular level of significance. This depends upon the usage of the batteries. Say they are to be used in pacemakers - in this case, the consequences of a type II error could be fatal (ie. the battery life is taken to be 500 days). Hence, we take α to be 0.1.
8. Now, the p-value (~0.01) is less than α (0.1), therefore, we might want to advise the end consumers to switch batteries with a safe margin in hand, such as once every year if the sample mean is around 480.