Chapter 20 – Testing Hypotheses about Proportions

1. Chapter 20 – Testing Hypotheses about Proportions

2. Hypothesis A hypothesis is: “a supposition; a proposition or principle which is supposed to be taken for granted, in order to draw a conclusion or inference for proof of the point in question; something not proved, but assumed for the purpose of argument.” From Webster’s Unabridged Dictionary, 1913

3. Testing claims about proportions • A company claims that it has a 90% satisfaction rate. • A politician claims that over 60% of the population supports his idea. • A restaurant claims that its burgers have less than 15% fat content. • A manufacturer claims that 98% of its products are defect-free. • A scientist claims that 13% of people are left-handed. • Coke claims more people like Coke than Pepsi. • Really saying more than 50% prefer Coke

4. Uncertainty in Hypothesis Testing We want to test the validity of a claim being made about the population. Just like with confidence intervals, we are going to use a sample, but this time to test the claim rather than come up with an estimate for the population value. Since we are dealing with just one sample, we still have sampling error to deal with. This means we can’t be 100% certain of our result, so we will have to state that uncertainty in our explanation.

5. Hypothesis testing is like a Court Case In a court case, what are the 2 possible verdicts against a defendant? Is the burden on the defense or the prosecution to make their case? How strong a case does it have to be?

6. Guilty or not guilty? • In court, the assumption is made that the defendant is not guilty and it is up to the prosecutor to show, beyond a shadow of a doubt, that the defendant is guilty. • We assume that the claim of the defendant is true. • We then have to see irrefutable evidence that the claim the defendant is making is false. • This is the same approach we use with hypothesis testing. • We reject the hypothesis only if evidence shows it to be unlikely to be true • We fail to reject the hypothesis if there isn’t enough evidence to show that it’s unlikely to be true

7. Hypotheses in Statistics • We have 2 types of hypotheses we use in statistics: • H0: the null hypothesis • HA: the alternative hypothesis • Null Hypothesis: • Always H0: p = # (in our text, might be different in others) • Represents the hypothesized value of our proportion in question • Assumed to be true unless proven false • We can either reject the null hypothesis • Or fail to reject the null hypothesis

8. Alternative Hypotheses • Depending on the type of claim being made we might have our alternative hypothesis be any of the following: • HA: p ≠ # - when claiming p is a certain value • HA: p > # - when claiming p is more than a certain value • HA: p < # - when claiming p is less than a certain value • The alternative hypothesis is the one we believe is likely to be true if the null hypothesis is shown to be unlikely to be true. (if we reject the null hypothesis)

9. Examples of Hypotheses: • A company claims that it has a 90% satisfaction rate. • H0: p = .9 • HA: p ≠.9 • A politician claims that over 60% of the population supports his idea. • H0: p = .6 • HA: p > .6 • A restaurant claims that its burgers have less than 15% fat content. • H0: p =.15 • HA: p < .15

10. P-Values At some point, we have to decide if the evidence is overwhelming enough to show that our defendant is now considered guilty. In statistics, we can quantify this using a P-value. We assume that the Null Hypothesis is correct and then look at our data. We need to find the probability that we would see data like ours if the Null Hypothesis is in fact true. This probability is our P-value.

11. Conclusions based on P-values When the P-value is very low, then it will be likely that our Null Hypothesis isn’t correct and should be rejected Otherwise, we can only say that there is not enough evidence to reject the Null Hypothesis, so we fail to reject it.

12. Model • Once we have our hypotheses determined, we need to decide which model we use to test them. • Always check conditions and assumptions. • Test about proportions involving one sample is called a one-proportion z-test. • Same assumptions and conditions as for one-proportion z-interval: • Independence • Randomization • 10% condition • Success/Failure condition

13. One-Proportion z-test • We test: • H0: p = p0 (p0 is the value in the claim) • Using and • When the conditions are met and the null hypothesis is true, this statistic z follows the standard Normal model, so we can use that to obtain a P-value.

14. Finding a P-value We use our test statistic to get our P-value. In this chapter, we are dealing with a one-proportion z-test. The P-value is supposed to measure the probability that we would see results like ours given that the null hypothesis is true.

15. More with P-values • We assume that the claim is correct and look at the probability we would get the result of our sample proportion by using the normal curve as we always have. • 3 Scenarios: • One-tailed: HA > value • One-tailed: HA < value • Two-tailed: HA≠ value

16. Examples HA < value HA≠ value

17. Finding a P-Value Example • Going back to our claim that the customer satisfaction rate is 90%. A sample of 100 customers show that 86% are satisfied. What’s the P-value? • H0: p = .9 • HA: p ≠.9

18. Conclusion • Conclusion is always about the Null Hypothesis • We need to state whether we reject or fail to reject the Null Hypothesis • If P-value is very low, we reject the Null Hypothesis • Otherwise, we fail to reject the Null Hypothesis • Conclusion should always include P-value

19. Another Example A politician claims that over 60% of the population supports his idea. A random sample of 250 potential voters shows that 152 support his idea. What can you conclude about this claim?