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Lecture 22.pdf

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Michael Krashinsky

Lecture Twenty Two ▯ Many times scientists and statisticians are interested in testing claims made about the mean of a distribution. ▯ The claim is called the null hypothesis of the test. Typically the claims or hypotheses of the test are written out in the following notation: H 0 ▯ = ▯ 0 null hypothesis H A ▯ 6= ▯ 0 alternative hypothesis ▯ When the alternative to the null hypothesis H : ▯ =0▯ is ▯ is 0igger or smaller than ▯ t0e test is said to be two sided. We will only consider two sided tests. ▯ To test that claim we sample from that distribution (ie collect data.) ▯ From the data we calculate the observed value of a statistic that we think will suggest evidence for rejecting the null hypothesis or suggest insu▯cient evidence for rejecting the null hypothesis. ▯ Given the role of the observed value of the statistic in putting the null hy- pothesis to the test, it is called the test statistic. ▯ If the probability of getting the test statistic that we observed when the claim is true is very small, then we think that we have evidence to reject the claim. ▯ The probability of getting what we observed when the claim is true is called the p-value of the test. ▯ In the situation where we reject the claim, it is possible that the claim is actually true and we just (by chance alone) observed an unusual and unlikely sample. ▯ We have to decide in advance the long run relative frequency we are willing to commit an error like that. ▯ The pre-decided relative frequency (in the long run) of committing an error like that, that we are willing to accept is called the level of signi▯cance of the test. 1 ▯ The steps involved in testing a null hypothesis are best understood through example. ▯ Example: The quantity of water in 750 ml bottles is only slightly non-normal. The standard deviation of the quantity of water in 750 ml bottles is 2 ml. I don’t want to get ripped o▯ but I don’t want water that I didn’t pay for either so I perform the test: H : ▯ = 750 0 H A ▯ 6= 750 Moreover, I decide in advance that (in the long run performing similar tests) I’m OK with incorrectly rejecting the null hypothesis only 5 percent of the time. Ie the level of signi▯cance of the test is decide to be ▯ = 0:05. 2 . 3 ▯ In testing the hypothesis H 0 ▯ = ▯ 0 H : ▯ 6= ▯ A 0 we are vulnerable to committing two types of mistakes: { Type I error, rejecting H when it’s true. 0 { Type 2 error, failing to reject H w0en it’s false. ▯ P(Type 1 error) = P(Reject H when 0t’s true) { Consider a test with level of signi▯cance ▯. { We reject H w0en p-value < ▯. { ▯ is the largest p-value we could get and still reject H . 0 { ▯ = P(Reject H whe0 it’s true) = P(Type 1 error). ▯ P(Type 2 error) = P(Fail to reject H when0it’s false) { Power = 1 ▯ P(Type 2 error) = 1 ▯ P(Fail to reject H wh0n it’s false) = P(Reject H when it’s false) 0 ▯ We want both of these small so we have to balance them. Another approach is to maximize the power of the test for a desired level of signi▯cance. ▯ Example: In the water bottle example above, what is the power of the test to detect a mean signi▯cantly di▯erent than ▯
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