Hypothesis Tests. w/ proportions. AP Statistics - Chapter 20

Hypothesis Tests w/ proportions AP Statistics - Chapter 20

let s say we flip a coin...

Let s flip a coin! # OF HEADS IN A ROW PROBABILITY 2 3 4 5 6 7 8 (0.5) 2 = 0.2500 (0.5) 3 = 0.1250 (0.5) 4 = 0.0625 (0.5) 5 = 0.03125 (0.5) 6 = 0.015625 (0.5) 7 = 0.0078125 etc. etc. etc.

1a. (the sampling bowl problem) It is believed that a sampling bowl in Ms. Calvin s classroom, filled with red and white beads, has 20% red beads in it. But, it is possible that someone has recently tampered with the bowl, replacing some of the red beads with white. Ms. Calvin takes a sample of 100 beads from the bowl, and 13 of them are red. a) Does this provide evidence that the bowl has a lower percentage of red beads than it used to? To answer this question, we need to know how likely it would be to get a sample where only 13/100 beads are red. We answer this type of question with a Hypothesis Test.

Hypothesis Tests here s our question: Could this observed change be the result of chance variation? or is the change so drastic (unlikely), that we consider it statistically significant?

Hypothesis Test Steps: 1. Identify your parameter of interest (p = or μ = ) 2. Conditions/Assumptions 3. Hypothesis statements 4. Calculations 5. Conclusion, in context

Conditions: Random sample? 10% rule? (only need to check when sampling without replacement) Success/failure: np 10 and nq 10 If the conditions are satisfied, we may use the Normal model to conduct a... 1-PROPORTION Z-TEST

HOW TO WRITE HYPOTHESES: Null hypothesis - A statement of no effect or no difference or no change H O : p = p O Alternative hypothesis - What we suspect is true or what we are trying to show H A : p > p O p < p O p p O

1a. (the sampling bowl problem) It is believed that a sampling bowl in Ms. Calvin s classroom, filled with red and white beads, has 20% red beads in it. But, it is possible that someone has recently tampered with the bowl, replacing some of the red beads with white. Ms. Calvin takes a sample of 100 beads from the bowl, and 13 of them are red. a) Does this provide evidence that the bowl has a lower percentage of red beads than it used to? p = true proportion of red beads in sampling bowl Ho: p = 0.2 Ha: p < 0.2 Conditions: - assume random sample of beads - assume 100 beads < 10% of all beads in bowl - np = 100(.2) = 20 and nq = 100(.8) = 80 Since both np and nq > 10, a normal model may be used

1a. (the sampling bowl problem), continued... It is believed that a sampling bowl in Ms. Calvin s classroom, filled with red and white beads, has 20% red beads in it. But, it is possible that someone has recently tampered with the bowl, replacing some of the red beads with white. Ms. Calvin takes a sample of 100 beads from the bowl, and 13 of them are red. a) Does this provide evidence that the bowl has a lower percentage of red beads than it used to? N(0.2, 0.04) Calculations for a hypothesis test means, find the probability of getting what you got in the sample. So we ll find a mean and standard deviation, and use a normal model to find that probability. z = 13 / 100-0.2 = -1.75 0.04 from z-table: 0.0401

1a. (the sampling bowl problem), continued... It is believed that a sampling bowl in Ms. Calvin s classroom, filled with red and white beads, has 20% red beads it. But, it is possible that someone has recently tampered Why did I find the probability LESS than 13/100? with the bowl, replacing some of the red beads with white. Ms. Calvin takes a sample of 100 beads from the bowl, and 13 of them are red. a) Does Since this my provide hypothesis evidence is that the p < bowl 0.2, I has want a lower to find percentage the probability of red of beads than getting it used 13/100 to? or MORE EXTREME in that less than direction. So I Calculations for a N(0.2, 0.04) find the probability of 13/100 or less. hypothesis test means, find the probability of getting what you got in z = 13 / 100-0.2 = -1.75 the sample. 0.04 So we ll find a mean and from z-table: 0.0401 standard deviation, and use a normal model to find that probability.

1a. (the sampling bowl problem), continued... It is believed that a sampling bowl in Ms. Calvin s classroom, filled with red and white beads, has 20% red beads in it. But, it is possible that someone has recently tampered with the bowl, replacing some of the red beads with white. Ms. Calvin takes a sample of 100 beads from the bowl, and 13 of them are red. a) Does this provide evidence that the bowl has a lower percentage of red beads than it used to? The using your calculator way!!! STAT -> TESTS -> 1-PropZTest p0 is your null hypothesis proportion (0.2 in this problem) x is the number of successes (13) n is the sample size (100) Then you must pick WHICH alternate hypothesis you want to test (on this problem, we are testing that the proportion is LESS than our p0 of 0.2, so pick the middle option, <p0). ***Make sure to write down: z = -1.75 AND p-value = 0.040059... the p= this screen gives you is the probability found from a normal model - we call it the p-value. (it is NOT the same as the p = true proportion you defined at the beginning of the problem.)

Level of Significance (alpha) α can be any value usual values: 0.10, 0.05, 0.01 (but 0.05 is MOST common) If we want a test that requires stronger evidence, we would use a LOWER alpha value.

Is our result statistically significant? If p-value α, reject the H O. If p-value > α, fail to reject the H O. Remember, α (level of significance) can be ANY value, but 0.05 is the most common. Typically, problems will tell you what level of significance to use. If not, use 0.05. NEVER accept the H O!!!!!

Writing your Conclusion: Since the p-value < (>) α, we reject (fail to reject) the H O. We have (lack) sufficient evidence to suggest that [H A in context]. Be sure to write H A in WORDS!

Interpreting your p-value (only need to do this when asked) Basically, you re explaining the probability you found on the normal model, in the context of the problem. This can be super tricky, but I tried to come up with a fill in the blank interpretation for you. If [H O is true], is the probability of getting a random sample whose statistic is [whatever it was] or [higher/lower/more extreme].

1a. (the sampling bowl problem), continued... It is believed that a sampling bowl in Ms. Calvin s classroom, filled with red and white beads, has 20% red beads in it. But, it is possible that someone has recently tampered with the bowl, replacing some of the red beads with white. Ms. Calvin takes a sample of 100 beads from the bowl, and 13 of them are red. a) Does this provide evidence that the bowl has a lower percentage of red beads than it used to? z = -1.75 AND p-value = 0.040059 Using α = 0.05 Since our p-value < α, we REJECT Ho. We HAVE sufficient evidence to suggest that the sampling bowl has lower than 20% red beads.

1b. (the sampling bowl problem) It is believed that a sampling bowl in Ms. Calvin s classroom, filled with red and white beads, has 20% red beads in it. But, it is possible that someone has recently tampered with the bowl, replacing some of the red beads with white. Ms. Calvin takes a sample of 100 beads from the bowl, and 13 of them are red. b) Carefully interpret the meaning of the p-value that was calculated in part (a). Assuming there really are 20% red beads in the bowl, 0.04 is the probability that a random sample would have 13 out of 100 red beads in it OR LESS.

2a. (the medical school problem) According to the Association of American Medical Colleges, only 46% of medical school applicants were admitted to a medical school in the fall of 2006. Two years later, a random sample of 180 medical school applicants was taken, and 77 were admitted to medical school. a) Does this data provide evidence of a change in the acceptance rate for medical Schools? Ho: p = 0.46 p = true proportion of applicants admitted to medical school Ha: p 0.46 Conditions: - given random sample of applicants - 180 likely < 10% of all med school applicants - np = 180(.46) = 82.8 > 10 and nq = 180(.54) = 97.2 > 10 We may use a normal model **For a not equal hypothesis, on the normal model, we re looking at BOTH tail end probabilities. We re looking for the probability of a sample proportion being THAT far off from the true mean, or more extreme - in EITHER direction. (see next slide)

2a. (the medical school problem) According to the Association of American Medical Colleges, only 46% of medical school applicants were admitted to a medical school in the fall of 2006. Two years later, a random sample of 180 medical school applicants was taken, and 77 were admitted to medical school. a) Does this data provide evidence of a change in the acceptance rate for medical Schools? So, we find the probability of the sample proportion being 77/180 or less and then we DOUBLE it. 0.1922 z = 77 / 180-0.46 = -0.87 z-table probability = 0.0371 DOUBLE IT!!! Using α = 0.05. Since our p-value (0.38) is greater p-value than α, we = fail 2(0.192) to reject = 0.3843 Ho! We lack sufficient evidence to suggest that the proportion of accepted med school applicants has changed.

2a. (the medical school problem) According to the Association of American Medical Colleges, only 46% of medical school applicants were admitted to a medical school in the fall of 2006. Two years later, a random sample of 180 medical school applicants was taken, and 77 were admitted to medical school. a) Does this data provide evidence of a change in the acceptance rate for medical Schools? On the calculator, though, we do the same thing we did before, just pick the p0 hypothesis option, and it ll do everything for you - including doubling the probability to get the p-value. See? The p-value is.3857, already doubled!

Using the Calculator & Showing Work: you may have noticed the p-values from the STAT->Tests menu are slightly different from the ones we calculated by hand using 1-PropZTest is actually MORE exact because it s not rounding anything mid-problem! HOWEVER if you are using 1-PropZTest to find your p-value, you still need to include the following as your work for the problem: the NAME of the test you are conducting (write 1-prop z-test somewhere) the z-score, p-value, and the alpha level you are using your conclusion (of course)

2b. (the medical school problem) According to the Association of American Medical Colleges, only 46% of medical school applicants were admitted to a medical school in the fall of 2006. Two years later, a random sample of 180 medical school applicants was taken, and 77 were admitted to medical school. b) Interpret the meaning of your p-value from part (a) in context. Assuming 46% of medical school applicants are accepted, the probability of a random sample having 77 out of 180 applicants (or a proportion more extreme) accepted is.3857..