Bell Work: A company with a fleet of 150 cars found that the emissions systems of 7 out of the 22 they tested failed to meet pollution control guidelines. Is there strong evidence that more than 20% of the fleet might be out of compliance? Test an appropriate hypothesis and state your conclusion. In 2001, a report indicated that about 3% of all births produce twins. Is the rate of twin births the same among very young mothers? Data from a large city hospital found that 7 sets of twins were born to 469 teenage girls. Test an appropriate hypothesis and state your conclusion.

Solutions: The conditions are not met – 22% is more than 10% of the population of 150 Births are independent but not random, so these teens may not represent the population. With a two-tailed test we get z -1.91 and a P-value of 0.0556, we may see a difference for these teens but it is not clear whether this can be generalized to all teenagers.

More About Tests and Intervals Chapter 21 Part 1

Example: Interpreting P-Values A researcher tested a new medication for allergies and concluded that it is more effective than an older medication. Explain what the P-value of 0.0015 means in this context. The chance of seeing an observed difference this large due to natural sampling variation is 0.15%.

The P-value is NOT the probability that the null hypothesis is true! Common Mistake: The P-value is NOT the probability that the null hypothesis is true!

P-value < 𝛼 P-value > 𝛼 Reject 𝐻 0 Fail to reject 𝐻 0 Alpha Levels (α) An alpha level or level of significance is a number we can compare the P-value to. It helps us decide whether to reject the null hypothesis or not. P-value < 𝛼 P-value > 𝛼 Reject 𝐻 0 Fail to reject 𝐻 0

The most common alpha levels are 0.10, 0.05, and 0.01 Alpha levels are chosen before performing the hypothesis testing.

Example: A one-proportion z-test was performed at 𝛼=0.05. It is found that the P-value is 0.0325. Based on this result, would you reject or fail to reject the null hypothesis? 0.0325 < 0.05, so I would reject the null hypothesis. There is sufficient evidence to support the alternative hypothesis.

Example: A one-proportion z-test was performed at 𝛼=0.01. It is found that the P-value is 0.0325. Based on this result, would you reject or fail to reject the null hypothesis? 0.0325 > 0.01, so I would fail to reject the null hypothesis. There is not sufficient evidence to support the alternative hypothesis.

Statistician Sir Ronald Fisher (another old white guy) came up with the alpha level of 0.05. In his book, The Design of Experiments, he discussed the amount of evidence needed to reject a null hypothesis. He said that while each problem was situation dependent, 1 out of 20 might be a reasonable value for scientific application. Since then most have treated the number 0.05 as the standard for hypothesis testing.

Today’s Assignment: To better understand what a P-value is, read pages 483-485 To better understand the whole chapter, READ THE CHAPTER! Ch 20 Practice Quiz HW: p.499 #1-6 Chapter 20 homework due today