Section 10.2 Tests of Significance

1. Section 10.2Tests of Significance AP Statistics March 2, 2010 Berkley High School, D1B1

2. Coin Flipping Example

3. Why did you doubt Mr. Fadoir’s truthfulness? • Because the outcome of the coin flipping experiment is very unlikely. • How unlikely? • .5^k, where k is the number of flips before you yelled.

4. Supposition (a fancy way of saying “unsupported”) • Built into the argument that “Mr. Fadoir is pulling our collective leg” is a supposition • What is that supposition? • “We suppose that the coin is fair.” • Where does the supposition show up? • .5^k

5. The Test of Significance • The test of significance asks the question: • “Does the statistic result from a real difference from the supposition” • or • Does the statistic result from just chance variation?”

6. Significance Test Procedure • Identify the population of interest and the parameter you want to draw conclusions about. State null and alternate hypotheses. • Choose the appropriate procedure. Verify the conditions for using the selected procedure. • If the conditions are met, carry out the inference procedure. • Calculate the test statistic. • Find the P-value • Interpret your results in the context of the problem

7. Example • Diet colas use artificial sweeteners to avoid sugar. These sweeteners gradually lose their sweetness over time. Manufacturers therefore test new colas for loss of sweetness before marketing them. Trained tasters sip the cola along with drinks of standard sweetness and score the cola on a “sweetness score” of 1 to 10. The cola is then stored for a month at high temperature to imitate the effect of four months’ storage. Each taster scores the cola again after storage. • What kind of experiment is this?

8. Example • Here’s the data: • 2.0, .4, .7, 2.0, -.4, 2.2, -1.3, 1.2, 1.1, 2.3 • Positive scores indicate a loss of sweetness. • Are these data good evidence that the cola lost sweetness in storage?

9. Significance Test Procedure • Step 1: Define the population and parameter of interest. State null and alternative hypotheses in words and symbols. • Population: Diet cola. • Parameter of interest: mean sweetness loss. • Suppose there is no sweetness loss (Nothing special going on). H0:µ=0. • You are trying to find if there was sweetness loss. Your alternate hypothesis is: Ha: µ>0.

10. Significance Test Procedure • Step 2: Choose the appropriate inference procedure. Verify the conditions for using the selected procedure. • We are going to use sample mean distribution: • Do the samples come from an SRS? • We don’t know. • Is the population at least ten times the sample size? • Yes. • Is the population normally distributed or is the sample size at least 25. • We don’t know if the population is normally distributed, and the sample is not big enough for CLT to come into play.

11. Significance Test Procedure • Step 3: Calculate the test static and the P-value. The P-value is the probability that our sample statistics is that extreme assuming that H0 is true. • µ=0, x-bar=1.02, σ=1 • Look at Ha to calculate “What is the probability of having a sample mean greater than 1.02?” • z=(1.02-0)/(1/root(10))=3.226, • P(Z>3.226) =.000619=normalcdf(3.226,1E99)

12. Significance Test Procedure • Step 4: Interpret the results in the context of the problem. • You reject H0 because the probability of having a sample mean of 1.02 is very small. We therefore accept the alternate hypothesis; we think the colas lost sweetness.

13. Assignment • Exercises 10.27-10.37 odd, 10.45-10.55 odd • Against All Odds Video www.learner.org, Episode 20.