Hypothesis testing:
A company which makes toy cars claims that its products have a mean life span of more than 5 years. Test the claim at 5% level of significance if a random sample of 28 toy cars was found to have a mean life span of 5.2 years with a standard deviation of 1.5 years.
Answers
Answer:
his is known as a proof by contradiction. You assume the opposite of your hypothesis is true and show that it can’t be true. If this happens, then your hypothesis must be true. All hypothesis tests go through the same process. Once you have the process down, then the concept is much easier. It is easier to see the process by looking at an example. Concepts that are needed will be detailed in this example.
Example 7.1.1 basics of hypothesis testing
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?
Solution
Well first, you should know what you are trying to measure. Define the random variable.
Let x = life of a XJ35 battery
Now you are not just trying to find different x values. You are trying to find what the true mean is. Since you are trying to find it, it must be unknown. You don’t think it is 500 days. If you did, you wouldn’t be doing any testing. The true mean, μ , is unknown. That means you should define that too.
Let μ = mean life of a XJ35 battery
Now what?
You may want to collect a sample. What kind of sample?
You could ask the manufacturers to give you batteries, but there is a chance that there could be some bias in the batteries they pick. To reduce the chance of bias, it is best to take a random sample.
How big should the sample be?
A sample of size 30 or more means that you can use the central limit theorem. Pick a sample of size 30.
Example 7.1.1 contains the data for the sample you collected:
491
485
503
492
282
490
489
495
497
487
493
480
483
504
501
486
478
492
482
502
485
503
497
500
488
475
478
490
487
486
Step-by-step explanation:
Now what should you do? Looking at the data set, you see some of the times are above 500 and some are below. But looking at all of the numbers is too difficult. It might be helpful to calculate the mean for this sample.
The sample mean is x¯¯¯=490 days. Looking at the sample mean, one might think that you are right. However, the standard deviation and the sample size also plays a role, so maybe you are wrong.
Before going any farther, it is time to formalize a few definitions.
You have a guess that the mean life of a battery is less than 500 days. This is opposed to what the manufacturer claims. There really are two hypotheses, which are just guesses here – the one that the manufacturer claims and the one that you believe. It is helpful to have names for them.