Therefore, the assumption that x is even is false and x is odd.

Transcription

1 Mathematical indirect proof: assume the opposite of the conclusion you wish to prove and show that with this supposition something must be false. Example: if an integer x squared, x 2, is odd then x is odd. indirect proof: we know that x 2 is odd assume that x is even (the opposite of x is odd)... and show some contradiction. If x is even then x =2n for some integer n. Then x 2 = (2n) 2 = 4n 2 = 2(2n 2 ).... and x 2 is even. This contradicts that x 2 is odd. Therefore, the assumption that x is even is false and x is odd.

2 Hypothesis (definition): a supposition; something not proved, but assumed for the purpose of argument; a proposition or principle which is taken for granted, in order to draw a conclusion or inference for proof of the point in question; Jurisprudence in the U.S. We assume a defendant is innocent. We must provide enough evidence to show the opposite. "enough": beyond a reasonable doubt O.J. Simpson criminal trial by the preponderance of the evidence O.J. Simpson civil trial criminal trial: "not guilty" no one said "innocent" civil trial: "guilty" "not guilty": The jury has not found the evidence strong enough to reject the defendant's presumption of innocence. The jury says "not guilty" and they do not say "innocent."

3 Back to Statistics, an example for this chapter. page 513 A large city's Department of Motor Vehicles claimed that 80% of candidates pass driving tests, but a newpaper reporter's survey of 90 randomly selected local teens who had taken the test found only 61 who passed. Question: Does this finding suggest that the passing rate for teenagers is lower than the DMV reported?

4 Population: all people who take driving tests. Sample: the sampled teenagers who took the driving test Proportion: the percent who pass the driving test p for the population p-hat for the sample We shall assume that the passing rate for teenagers is the same as the DMV's overall rate of 80%, unless there is strong evidence that it is lower. The null hypothesis H 0 : p = 0.80 The alternative hypothesis H A : p < 0.80 Can we use a Normal model for calculating: 0) independence a) random sample b) 10% rule c) np and nq >= 10 YES, check them out.

5 The P-value is the probability that any sample's proportion is 61/90 (or worse). Let's calculate, using Normal model: Pr( p-hat < 61/90) convert to z-scores... = Pr( ) 0.002

6 CONCLUSION (text page 515) Because the P-value of is very low, I reject the null hypothesis. These survey data provide strong evidence that the passing rate for teenagers taking the driving test is lower than 80%. If the passing rate for teenage driving candidates were actually 80%, we'd expect to see success rates this low in only about 1 in 500 samples (0.2%). This seems quite unlikely, casting doubt that the DMV's stated success rate applies to teens.

7 Second example for this chapter. page 516 Advances in medical care such as prenatal ultrasound examination now make it possible to determine a clild's sex early in a pregnancy. There is a fear that in some cultures some parents may use this technology to select the sex of their children. A study from Punab, India... reports that, in 1993, in one hospital 56.9% of the 550 live births that year were boys. It is a medical fact male babies are slightly more common than female babies. The study's authors report a baseline for this region of 51.7% male live births. Question: Is there evidence that the proportion of male births has changed? WARNING: this is a very different situation than the last one. the last example we checked if p-hat was too small. this example we need to check if p-hat is different, not equal.

8 Population: all babies born in that one hospital in India. Sample: the sampled 550 live births that year. Proportion: the percent of babies born who are male p for the population p-hat for the sample We shall assume that the proportion of male births is 51.7% The null hypothesis H 0 : p = The alternative hypothesis H A : p <> 0.517, i.e. not equal Can we use a Normal model for calculating: 0) independence a) random sample b) 10% rule c) np and nq >= 10 YES, check them out.

9 The P-value is the probability that any sample's proportion is far from in either direction. Let's calculate, using Normal model: P-value = 2*Pr( p-hat > 56.7%) convert to z-scores... = 2*Pr( )

10 CONCLUSION (text page 518) The P-value of says that if the true proportion of male babies were still at 51.7% then an observed proportion as different as 56.9% male babies would occur at random only about 15 times in With a P-value this small, I reject H 0. This is strong evidence that the birth ratio of boys to girls is not equal to its natural level. It appears that the proportion of boys may have increased.