# Hypothesis testing

I have a question regarding how to formulate the null and alternative hypothesis for two questions. It appears that they would both require a “greater than” hypothesis yet one does and one does not. Could anyone shed some light on this for me? Thank you. Question 1: You have looked at ABC capital’s fund which has been in existence for 24 months. During this time, its mean monthly return has been 1.5% with a monthly sample standard deviation of 3.6%. Given the market risk and according to the pricing model, it was expected to have earned 1.1% mean monthly return. Assuming the results are normally distributed, are the actual results consistent with an underlying or population mean monthly return of 1.1%? I thought this would have required:** H(0) 1.1% **** but instead the book says that:**H(0) = 1.1% and H(a) =/ 1.1%. Question 2: SuperClothes is concerned about a slowdown in payments from vendors. The rate of payment is measured by the average number of days in receivables. SuperClothes has historically an average of 45 days in receivables. They use a sample to track vendor payment rates. A random sample of 50 accounts shows a mean number of days in receivables of 49 and a standard deviation of 8 days. The answer to this question says that because the suspected condition (49 days) is greater than the historical number of days (45 days), this is a “greater than” hypothesis. But how is this different from question 1 where the historical mean (1.1%) is less than the actual mean (1.5%) but there it was a “not equal to” hypothesis?

Question 1 key phrase: Assuming the results are normally distributed, are the actual results consistent with an underlying or population mean monthly return of 1.1%?

They’re asking if the sample provides evidence that the true (population) mean monthly return differs from 1.1%.

In other words, is mu equal to 1.1% or is mu not equal to 1.1%? The sample mean is irrelevant to determining H0 and Ha.

Question 2 key phrase: _ SuperClothes is concerned about a slowdown in payments from vendors. _ The rate of payment is measured by the average number of days in receivables. SuperClothes has historically an average of 45 days in receivables.

A slowdown in payments collection would mean an increase in the average number of days in receivables. The historical average (they’re assuming this is the true average) has been 45 days. If they want to determine if the true mean (mu) has increased, they would test if mu is greater than 45 days. The null is that mu is less than or equal to 45 days, and the alternative hypothesis is that mu is greater than 45 days. Again, the sample mean of 49 is not needed when determining the null and alternative hypotheses. The key phrase, as I mentioned, is the slowdown in collection and the increase of days receivables over 45 days.

Hope this helps!

Thank you. That was very helpful. I have one more question based on this topic, though.

A study has the following results:

“ABC discovered that defaulted bonds for the petro-chemical industry had higher recvery rates than bonds for other industries. Could the difference in the recovery rates be caused by a larger percentage of senior bonds in the petro-chemical sector?”

With this question, though the actual results were demonstrated to be greater than the supposed results, they didn’t set out with that in mind. For example they never asked the question: "Are petro-chemical bonds more likely to have good recovery rates versus other industry bonds?"

Thanks!

For this: “ABC discovered that defaulted bonds for the petro-chemical industry had higher recvery rates than bonds for other industries. Could the difference in the recovery rates be caused by a larger percentage of senior bonds in the petro-chemical sector?”

I might not be understanding what you’re asking, so let me know if I missed it.

It seems the researchers sampled the populations, and the calculated statistics showed that the recovery rate was highest for the PC industry when compared with others.

It doesn’t mention whether this is a statistically significant result, though. For example, they could have conducted a statistical test under the null hypothesis of equality of recovery rates. Upon finding evidence of differences in the rates for the industries, they could (should, probably) have followed up with further tests to rank these recovery rates (find high, middle, low, etc., thus answering the question of “which industry has the highest/lowest recovery rate?”) .

But again, they don’t mention this, and it could be as simple as the researchers calculating some statistics without conducting any tests.

In either case, it doesn’t appear that they set out with the idea that PC bonds had higher recovery rates (I think your main question).

Let me know if that was what you’re looking for here.