AP Statistics Chapter 20 Notes “Hypotheses Testing for Proportions”

Rejecting or Retaining Hypotheses A hypothesis is an unproven theory. It is an idea that we tentatively hold on to while we test our theory. If we keep the hypothesis after our test is complete, we retain it. If we throw out the hypothesis after our test is complete, we reject it. Life examples: Science Fair Projects Jury Duty – (innocence is the starting hypothesis until it is rejected and guilt is decided)

Hypotheses Testing in Statistics The starting hypothesis is called the null hypothesis (HO). You will test the null hypothesis and then use your results to either retain it or reject it. Every statistics problem has an alternative hypothesis (HA). If you reject the null hypothesis, then you are accepting the alternative as true. RETAINING THE NULL HYPOTHESIS DOES NOT MEAN IT IS ACCEPTED AS TRUTH. IT MEANS WE FAILED TO REJECT IT BASED ON THE EVIDENCE FROM OUR SAMPLE.

Example – Is there a home field advantage? In the 2002 major league baseball season, 2425 regular season games were played and the home team won 1314 times, or 54.2%. If there were no home field advantage, the home team would win about 50% of the games played. Is this deviation from 50% explained just from natural sampling variability or is this evidence to suggest the presence of a “home field advantage” in professional baseball?

Step 1: State the Hypotheses. (HO and HA) If there is no home field advantage, then p would be 50%. So HO : p = 50% HA will be p > 50%, because the alternative we are looking for is that the home team wins more often.

Step 2: Check the Conditions 1. SRS – We assume the 2002 season was randomly selected from all seasons of baseball. 2. Independence – We assume the outcome of one game does not influence the outcome of another game. (No revenge factor) 3. Population Size – The sample size is 2425 games. There have been more than ten times that or 24,250 professional baseball games ever played so this condition has been met. 4. Minimum of 10 successes (np>10) and 10 failures (nq>10) – There were 1314 games where the home team won and 1111 games where the home team lost in this sample so this condition has been met.

Step 3: Do the Calculations STAT TESTS #5. 1 – PropZTest Po = the null hypothesis = .50 x = the number of successes = 1314 n = the sample size = 2425 Choose >Po b/c the alternate hypothesis is that you think p is higher than the null hypothesis Mean: SD: Z-score: 4.1223 Probability (“p-value”): 1.877E-5 = almost 0% HO : p = 50% This probability is the probability of your sample’s results occurring naturally. Since this probability is almost 0%, there is almost no chance of our sample’s results occurring naturally. Therefore, the null hypothesis is not true and it should be rejected.

Step 4: State your Conclusion (Reject or Retain the Null Hypothesis) If the probability (“p-value”) is smaller than 5%, we reject the null hypothesis. (We reject because there is a less than 5% chance that our sample’s results could be just natural sampling variability.) SAY THIS: “We reject the null hypothesis of p = 0.50. There is sufficient evidence from our sample to conclude that there is a presence of a home field advantage.” If the probability (“p-value”) is larger than 5%, we retain the null hypothesis. (We retain because there is more than a 5% chance that our sample’s results could be just natural sampling variability.) SAY THIS: “We retain the null hypothesis that p = 0.50. There is insufficient evidence from our sample to conclude that there is a presence of a home field advantage.”

Ho and Ha Ho is always: p = Ho HA can be: p > Ho (You think the true % is higher than the null hypothesis) p < Ho (You think the true % is lower than the null hypothesis) p = Ho (You think the true % is different than the null hypothesis) .

Another Example It’s a medical fact that male babies are slightly more common than female babies. We want to know whether the proportion of male births has changed from the established baseline of 51.7%. To test this hypothesis, a study in India reported in 1993 that 56.9% of one hospital’s 550 births that year were boys. Could this variation be explained by natural sampling variability or is there evidence to suggest that there is really a change in the percentage of male births?

The Process of Hypothesis Testing Step 1: State the hypotheses. Step 2: Check the conditions. Step 3: Do the Calculations. Step 4: State your Conclusion. HO : p = 51.7% STAT, TESTS, #5 Po =.517 x = .569*550 = 313 (round) n = 550 Choose = Po b/c the alternate hypothesis is that you think p has changed from the null hypothesis Ha : p = 51.7% - We assume this year was chosen randomly among all birth records at this hospital and that each birth is independent of the other births. - We assume that there have been at least 5,500 births at this hospital. The probability of your sample’s results occurring naturally (the p-value) is 1.45%. Since this probability is less than 5%, there is very little chance of our sample’s results occurring naturally. Therefore, the null hypothesis is not true and it should be rejected. Our sample indicates that the percentage of male births has changed. - We have at least 10 male births and 10 female births in this sample.

Using A Confidence Interval to Test a Hypothesis Find a 95% confidence interval for p based on your sample proportion ( ) in the last example. Is the null hypothesis (p = .517) contained in this interval? Does that mean we should reject or retain the null hypothesis? STAT, TESTS, A x = .569*550 = 313 (round) n = 550 C-level = .95 52.771% to 61.048% No, the null hypothesis is outside of the confidence interval We should therefore reject the null hypothesis