This week I received a question from the ASQ Librarian concerning a person’s question about one of the CRE Question bank questions. It was a nice twopart question concerning a hypothesis test of a sample means value and degradation.
Here’s the question as sent over for consideration.
Five samples of a product are tested for power after a reliability testing to assess any degradation. The products measured 10, 10.1, 10.5, 9.8, and 10.7 watts. If the average wattage is greater than or equal to 10 watts at 95% confidence level, we can conclude that the degradation has occurred. Calculate the appropriate statistics and conclude  
A. 
 
B. 
 
C. 
 
D. 

From the question, it is clear that we want to compare the sample mean value to the given 10 watts average and determine if the sample shows evidence that degradation has occurred. The problem says that degradation is indicated with higher values of wattage, over 10 watts.
Degradation is commonly thought to only reduce values like brake pad wear will show a decrease in pad thickness over time as it wears. Depending on the measure and system degradation may result in an increasing value over time. For example it may take more current, A, to start a fan motor to overcome initial friction as the fan bearings wear and/or the bearing lubrication deteriorate. We can measure motor current without taking apart the fan compared to bearing dimension or lubrication quality with require the deconstruction of the fan.
The person asking the question about this question felt that degradation is smaller is worse which is the reverse needed in for this situation.
Back to the question and how to solve it. First, set up the hypothesis test, we are interested in detecting if there is evidence of degradation and values higher than 10 watts on average. Thus the null hypothesis is if there is no degradation and the alternative hypothesis is higher than 10 watts.
$$ \large\displaystyle \begin{array}{l}{{H}_{0}}:\mu < 10\text{ watts}\\{{H}_{1}}:\mu \ge 10\text{ watts}\end{array}$$
We will need the degrees of freedom to enter the ttable to find the critical value at 95% confidence. With n = 5 samples, DF = (n1) = (51) = 4. We also know we are interested in the right tail of the t distribution as we are seeking a greater than the mean value, onesided, situation.
The critical value of the t – table at 0.95 right tail is 2.132. Thus, if the test statistic is greater than this value, we conclude the alternative hypothesis is true. If it is less, we do not have convincing evidence of degradation.
The next step is to calculate the test statistic.
$$ \large\displaystyle \begin{array}{l}t=\frac{\bar{X}\mu }{{}^{s}\!\!\diagup\!\!{}_{\sqrt{n}}\;}\\=\frac{10.2210}{{}^{0.37}\!\!\diagup\!\!{}_{\sqrt{5}}\;}\\=\frac{0.22}{0.165469}\\=1.329\end{array}$$
Finally we compare the test statistic with the critical value, 1.329 is less then the critical value of 2.132, thus we stay with the null hypothesis that there is no evidence of degradation.
For the questions answer the statistic is 1.329 and No degradation, thus select response A.
How would you approach this problem? Anyone have a better example of degradation revealed with an increasing measure?
Related:
PairedComparison Hypothesis Tests (article)
Hypothesis Tests for Proportion (article)
Mann Reverse Arrangement Test (article)
Ernest Paul Jones says
I am a confirm believer of your illustration because it has a sound analysis as well as explanation. But, I was actually going to assume the same answer by eye balled inspection of the data. I saw that only one element of data showed to be below 10. by assuming that all of the readings were related to degeneration it is accurate to conclude that degeneration occurs above the stated mean.
But, I still wanted to see the valid proof.