Chapter 20 Testing Hypotheses About Proportions

confidence intervals and hypothesis tests go hand in hand:  A confidence interval shows us the range of plausible values for p.  A hypothesis test makes a yes/no decision about a plausible value for p.

A Trial as a Hypothesis Test – “INNOCENT UNTIL PROVEN GUILTY!” – If we have reasonable doubt… – We reject the hypothesis of innocence and declare the person guilty.

The Reasoning of Hypothesis Testing 1. Hypotheses 2. Conditions 3. Mechanics 4. Conclusion

1.Hypotheses – The null hypothesis: The hypothesis we assume true. H 0 : parameter = hypothesized value. – The alternative hypothesis: H A, (if we reject the null): H A : parameter < hypothesized value H A : parameter ≠ hypothesized value H A : parameter > hypothesized value

2.Conditions -Randomization -10% condition -Success/Failure -Large Enough

3.Mechanics (calculations) 1. Find/Identify 2. calculate the z-score for using: 3. Find the probability asked in the problem using normalcdf(lower,upper) (this prob. is called a p-value) (H 0 : p = p 0 )

3.Mechanics GOAL: obtain a P-value. The P-value is the probability of the observed data happening if the null hypothesis were correct.

4.Conclusion – either…. – we reject the null (when p-value less 0.05), – or we fail to reject (not accept!) the null hypothesis (when p-value greater than 0.05).  “fail to reject the null hypothesis” when the data are consistent with the null hypothesis  “reject the null hypothesis,” when data would be very unlikely were the null true.

What to Do with an “Innocent” Defendant If the evidence is not strong enough to reject innocence, the jury returns with a verdict of “not guilty.” – The jury does not say that the defendant is innocent. – All it says is that there is not enough evidence to convict, to reject innocence. – The defendant may, in fact, be innocent, but the jury has no way to be sure.

Said statistically, we will fail to reject the null hypothesis. – We never declare the null hypothesis to be true, because we simply do not know whether it’s true or not.

In a trial, the burden of proof is on the prosecution. In a hypothesis test, the burden of proof is on the unusual claim. The null hypothesis is the ordinary state of affairs, so it’s the alternative to the null hypothesis that we consider unusual (and for which we must marshal evidence).

**Don’t accept the null hypothesis. **Don’t make your null hypothesis what you want to show to be true.

Summary of Hypothesis Testing: – Start with a null hypothesis. – Alternative hypothesis can be one- or two- sided. – Check assumptions and conditions. – Data are out of line with H 0, small P-value, reject the null hypothesis. – Data are consistent with H 0, large P-value, don’t reject the null hypothesis. – State the conclusion in the context of the original question.

Chapter 20 Assignment Pg. 476: #1, 9, 11, 13, 17, 21, 25, 27