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Hypothesis Testing

Hypothesis

The formal testing of hypothesis is a critical step not to be ignored. One of the jobs of statistics is to reduce the uncertainty in our decision making. It does so by helping to identify assumptions and check the validity, in essence, what is really there. It is my personal preference to evaluate stocks or mutual funds to see if they actually exceed the average market return of 10%. The 10% return is the historical return on stocks over the past 75 years, however, each year has its own unique return. When doing so, I am interested in did this mutual fund exceed the 10% annual return. When you look at hypothesis testing there are three ways you can set them up. There are as follows:

HO: = H1: not equal Word In story problem: Is Different

Ho: < or = H1: is greater than Word In story problem: more than, increased

HO: > or = H1: Is less than Word In story problem: decreased, dropped

The hypothesis come in “pre-packaged sets” so you can’t mix and match. So when you are setting them up, look at the alternative (H1) in setting them up. Many times this is the decision that interests you the most. In this case, I wanted returns that exceeded ten percent, so I looked at the alternative hypothesis of greater than when setting up the process. What is helpful in thinking about the hypothesis and alternative hypothesis are the unique qualities of the null or HO, hypotheses. The HO hypothesis has unique characteristics. The unique characteristics is that you assume there to either be “no difference” or status quo. So, I set the hypothesis up as follows:

HO: return is less than or equal to ten percent

H1: return is greater than ten percent

For the hypothesis test, I could have chosen to have it set up two other ways and those are:

HO: the return is equal to ten percent

H1; the return is not equal to ten percent

HO: The return is greater than or equal to ten percent

H1: The return is less than ten percent.

What is helpful in thinking about the hypothesis and alternative hypothesis are the unique qualities of the null or HO, hypotheses. The HO hypothesis has unique characteristics. The unique characteristics are that you assume there to either “no difference” or status quo. In the stating of HO there is always an implied equal. To help you understand this concept lets take a trip to the local vending machine.

Hypothesis Test Thinking and a Vending Machine

Imagine yourself walking up to a vending machine. You want to purchase a bottle of Pepsi for a $1.00. You walk up to the vending machine and assume you are going to put your dollar in the machine, press the Pepsi button, and the machine will dispense the bottle of Pepsi. Do you walk up to the machine, knowing that it is not going to give you your bottle of Pepsi after you insert your dollar? If so, would you proceed with the transaction. No you wouldn’t make the transaction and willingly lose your dollar. You assume that as you approach the machine and put your money in that you will receive your Pepsi because that is what has happened time after time, this is the “status quo” segment of the hypothesis. Now, when the machine doesn’t give you your Pepsi but keeps your dollar, you REJECT the hypothesis that this machine is fair and ACCEPT the alternative hypotheses that this machine has stolen my money and is a crooked machine. Now that you know about the formal hypothesis, let’s look at the rest of the procedure to separate fact from fiction.

Remaining Hypothesis Testing Steps

Decision Rule:

The remaining steps are to set up the rules and check our results. We set our rules up ahead of time so as not to be unduly influenced by the results. So, we decide on a level of significance. This is a significant level because it determines the point at which we either accept or reject our null hypothesis. The levels are normally.10, .05 or .01. These are similar to purchasing a Pepsi in a small, medium or large, sorry; we don’t have a super size in statistics. When choosing the levels of significance, one can look at industry norm or you can choose any one of the three. Normally, the .05 level of significance is chosen because this is middle ground. So we chose the .05 level of significance.

Test Statistic

The next step is to calculate our “test statistic”. This is the statistic that our computer software generates for us. This is most easily expressed in probability or what is normally called the “P” value. The “P” stands for probability and can be thought of this way. If the “true average”, the average that is really there, not our sample results, the probability of getting a sample average which we found, is some value. Lets say that we were testing the hypothesis that our mutual fund return is less than or equal to ten percent and we input our data, test the hypothesis at the .05 level of significance. The computer software calculates a “P value” of .15. A .15 value is greater than the .05 level of significance and so we don’t reject our hypothesis and conclude the stock we are looking at has a return less than or equal to 10%. However, if the “ P value” calculated from the computer program was .04; we would have rejected the null hypothesis and concluded that the average return of this mutual fund was greater than ten percent.

Side Note: if the “P or probability value calculated “is less than the level of significance, we reject the null hypothesis and accept the alternative. We do this because we are saying that if the null hypothesis is true, the probability of getting a sample average with this result is less than the decision rule statistic.

Decision:

The decision is where you compare the probability value calculated against the critical probability value set in your decision rule. If your test probability value is less than the critical, reject the null hypothesis and accept the alternative hypothesis. The P value testing allows for you to be slightly confident, if is just on the border line of accept or reject to very confident if the probability value is very small such as .001 (1/1000)

You may need to double check your values and if unsure, you can ask yourself do I want a 5% raise or do I want a .1% raise. This is a simple way of keeping your decision correct and not confusing the decision because you misinterpreted the p values size.

Traditional Approach

In 2004, a small dealership leased 21 Chevrolet Impalas on two year leases. When the cars were returned in 2006, the mileage recorded was (see table). Is the dealers mean significantly greater than the national average of 30,000 miles for two year leased vehicles at the 10% level of significance?

T value = 21-1 = 20 , one tailed test, .10 level of significance = 1.325

HO: The average miles is less than or equal to 30,000

H1: The average miles are greater than 30,000

Decision Rule: If the T value calculated is greater than 1.325, reject HO

Test Statistic: T= ( 33,950-30,000)/ (11,866/√21)

= ( 33,950-30,000)/ (11,866/4.58)

= 3,950/ 2,590

T value =1.52

Decision: Since the Test statistic of 1.52 is greater than the T critical value of 1.325, we reject HO and conclude the cars have more mileage.