Introduction to Hypothesis Testing

11/4/2008 Hypothesis Testing Introduction to Hypothesis Testing Errors in Decision Making
Do juries ever make mistakes? 1. If a person is really innocent, but the jury decides (s)he’s guilty, then they've sent an innocent person to jail. Type I error. 2. If a person is really guilty, but the jury finds him/her not guilty, a criminal is walking free on the streets. Type II error.
Joe didn’t want to go to court right away to sue the Rainbow company, so he decided to put them to a test. The Cloud 9 Club randomly selected 1000 parachutes from the Rainbow company, and tested each and every one of them.
The main parachutes failed to open for16 of them.
“That’s it. 16 is just too many. That’s a failure rate of 1.6%. ”, said Joe. “I will sue Rainbow company because it appears that their main parachute failure rate is more than 1%. They should pay for my injured members’ medical bills.”
Hypothesis testing in science is a lot like the criminal court system in the United States. How do we decide guilt? 1. Assume innocence until “proven” guilty. 2. Evidence is presented at a trial. 3. Proof has to be “beyond a reasonable doubt”
A jury's possible decision:
guilty
not guilty
Note that a jury cannot declare somebody “innocent”, just “not guilty”. This is an important point. A hypothetical case
Joe Flyer is the owner of the Cloud Nine Skydiving Club. The members in the club use parachutes manufactured by the Rainbow company. Many of the members have complained that the main parachutes fail to open too often. Last week one of the members suffered a hip fracture and a spinal injury due to the failing main parachute.
Joe’s lawyer contacted the parachute company, and they claimed that their main parachute failure rate is only 1%, which is less than their competitor’s failure rate, so no one should complain about Rainbow parachutes. Meet the opponents Hi, I'm Joe Flyer. Hi, I'm Greg Walton I represent my company, Rainbow Parachutes. 1 11/4/2008 In the courtroom Claims
ALL RISE. The Honorable Judge Just presiding.
That is,
1. State your claims: I challenge your company's claim. I claim that MORE than 1% of your main parachutes fail. We maintain our claim that our main parachute failure rate is 1%. That is, the proportion of failing main parachutes IS only 0.01. Wait a minute! We have to assume innocence until “proven” guilty. Do you have any evidence to support your claim? Your parachutes are really bad. We tested them! You should pay for our medical bills!
The null hypothesis is H0: p = 0.01
The alternative hypothesis is Ha: p >0.01 OK, so let’s assume he’s correct and the proportion of defective parachutes is 1% Let’s find the probability that out of 1000 randomly selected parachutes 16 will fail to open. Imagine that we take many, many, many random samples of size 1000 from all your parachutes and find the proportion that fail in each sample. The sampling distribution of the sample proportion The sampling distribution of the sample proportion
The evidence comes from the sampling distribution of these sample proportions.
According to statistical theory, the sampling distribution of the sample proportion is approximately normal since np = 1000(0.01) = 10 and n(1-p) = 1000(0.99) = 990 are both large enough. The mean of the sampling distribution is p = 0.01, and the standard deviation is p(1 − p) = n 0.01 ⋅ (0.99) ≈ 0.00315 1000 σ p$ = p(1 − p) ≈ 0.00315 n µ p$ = p = 0.01 2 11/4/2008 Significance level Oh, I am sure that a 1.6% failure rate would happen just by chance at least 510% of the time. Using the sampling distribution with your CLAIMED value of p=1%, let’s see what’s the probability of a failure rate as high as 1.6% is. I have to state, gentlemen, that in this court we usually use a 5% significance level. That means, if the observed value could happen by chance more than 5% of the time, then the evidence is not strong enough, and the verdict will be “not guilty.” So let’s see. The evidence
Our sample proportion is p$ = 16 = 0.016 1000
To find the probability of such a failure rate we first need to find the z-score for this sample proportion (still assuming innocence!): z= p$ − p 0.016 − 0.01 = ≈ 1905 . 0.00315 p(1 − p) 1000 0.028 z = 1.905
Let’s find the upper tail probability, which we will call the p-value. Conclusion AHA! The p-value is 0.028. That is less than 3%. The verdict Uh-oh, we are in deep trouble…3% is less than 5%.... Yes, the p-value is less than our significance level, α =5%. That means the evidence is strong enough to say beyond reasonable doubt that we can reject the Rainbow company’s claim that the failure rate is 1%. The Rainbow company is found to be guilty, and so they need to pay for the medical bills of the injured, and recall your products. 3 11/4/2008 Case closed 4