Section 10.2 Tests of Significance AP Statistics April 2013 Section 10.2 Tests of Significance

Coin Flipping Example On a scrap paper record the results of my coin flips.

Why did you doubt my 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.

Supposition (a fancy way of saying “unsupported”) Built into the argument that “Mrs. Mullen 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

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?”

Example: I claim that I make 80% of my free throw shots. To test my claim, you ask me to take 20 free throw shots. I make only 8 out of 20 You respond, “I don’t believe your claim. It is unlikely that an 80% shooter would make only 8 out of 20.

Significance Test Procedure STEP 1: Define the population and parameter of interest. State the null and alternative hypotheses in words and symbols. Population: My free throw shots Parameter of interest: Proportion of shots made Suppose I am an 80% free throw shooter. This is a hypothesis and we think it is false. We will call it the null hypothesis and use the symbol H0 (pronounced H - nought). H0 : p = 0.8 You are trying to show that I am worse than an 80% shooter. Your alternate hypothesis is: Ha : p < .08

Significance Test Procedure STEP 2: Choose the appropriate inference procedure. Verify the conditions for using the selected procedure. We are going to use the Binomial Distribution Each trial has either a success or a failure There is a set number of trials Trials are independent The probability of a success is constant

Significance Test Procedure STEP 3: Calculate the P-value. The P-value is the probability that our sample statistic is that extreme assuming that H0 is true. Look at Ha to calculate “What is the probability of making 8 or fewer shots out of 20?” Binomcdf ( 20, .8, 8) = .000102

Significance Test Procedure STEP 4: Interpret the results in the context of the problem. You reject H0 because the probability of being an 80% shooter and making only 8 out of 20 shots is extremely low. You conclude that Ha is correct; the true proportion is < 80% There are only two possibilities at this step: You reject H0 because the probability is so low. We accept Ha You fail to reject H0 because the probability is not low enough

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

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?

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?

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.

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 30. We don’t know if the population is normally distributed, and the sample is not big enough for CLT to come into play. However, the book did say “from long experience we know that individual tasters’ scores vary according to a normal distribution.”

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)

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.

Limits of z-Test There is a major limitation to the z-Test, and that is it can be used only if both μ and σ are known, which is unlikely.

Limits of z-Test In cases where the population mean is known, but the population standard deviation is not, if a sample is sufficiently large, it is acceptable to use the estimated population standard deviation in place of the true population standard deviation to calculate an estimated standard error of the mean. However, there is no 'real' criterion for how large a sample must be before it is “sufficiently” large.

Assignment Exercises 10.29-10.40 all.