Chapter 8: Hypothesis Testing for Population Proportions

Chapter 8: Hypothesis Testing for Population Proportions The basics of Significance Testing

Statistical Inference • Already discussed confidence intervals for unknown population parameter, p • Confidence Intervals used when the goal is to estimate an unknown population parameter like ρ (like when we estimated the true proportion of all 5,000 COC students who have at least one tattoo) • This chapter... statistical inference through significance tests • Evaluate evidence (a statistic) provided by sample data about some claim concerning an unknown population parameter like ρ

The Main Ingredients of Hypothesis Testing • There once were four students who missed the midterm for their statistics class. They went to the professor together and said, “Please let us make up the exam. We carpool together, and on our way to the exam, we got a flat tire. That’s why we missed the exam.” The professor didn’t believe them, but instead of arguing he said, “Sure, you can make up the exam. Be in my office tomorrow at 8.”

The Main Ingredients of Hypothesis Testing • The next day, they met in his office. He sent each student to a separate room and gave them an exam. The exam consisted of only one question: “Which tire?” • Let’s image all four students answered, “left rear tire.” • So... what do you think? Were students most likely telling the truth? Lying?

Let’s assume the students were lying... • What are the chances of all of them guessing the same tire? • Let’s simulate; using Stat Crunch, input RFront, LFront, RRear, LRear • Data, sample, choose your data, sample size 4, number of samples 10, sample with replacement • How many times, just by random chance, does Stat Crunch choose the same tire? Let’s create a dot plot on the board; what do you think?

Let’s assume the students were lying... • Assuming the students were lying, the chances that all four of them would guess the same tire, just by random chance, according to our simulation, is ... look at our dot plot... • If we carried out this simulation again, would we get the same data? The same exact dot plot?

The Main Ingredients of Hypothesis Testing • Surprised or not? • The professor suspected they had been lying. That’s why he did what he did. • Maybe they just got lucky ... just by chance they all guess the same tire. How ‘lucky’ would they have to be? • The theoretical probability that all four students would guess the same tire is about ... • Do you consider that likely/typical or unlikely/rare that they could have just simply, by chance guessed the same tire? Look at our dot plot...

I’m a great free-throw shooter... • Let’s think about another hypothesis test/inference example/situation...

I’m a great free-throw shooter... • I claim that in the last 5 years of playing basketball, I, on average, make 90% of my basketball free throws. • To test my claim, I am asked to shoot 10 free throws. I make 2 of the 10 (only 20%). • Do you still believe my claim that I make 90% of my free throw? Why or why not?

I claim 90% ... I actually made only 20%... • Do you still believe my claim that I make 90% of my free throw? Why or why not? • Do you agree that statistics vary from sample to sample? So if I attempted another 10 free throws, chances are I would make something other than 2 of them? • So the question is... would me actually making only 2 out of 10 (or 20%) happen so, so very rarely (assuming my 90% claim were true) that now you are starting to question/doubt if my claim really is true.

I claim 90% ... I actually made only 20%... • Let’s simulate. Let’s assume that we believe the claim. Pull up the random digits table. Let’s say 0 through 8 represent making a basket; 9 represents missing a basket. • Go into the table on a random line, look at ten 1-digit numbers, duplicates are OK. • Count the number of ‘baskets’ made in ten 1-digit numbers. Do this five times. Put your magnets up on the class dot plot. • So the question is... would me actually making only 2 out of 10 (or 20%) happen so, so very rarely (assuming my 90% claim were true) that now you are starting to question/doubt if my claim really is true.

Hypothesis Testing or Significance Testing ... • A formal procedure that enables us to choose between two hypotheses when we are uncertain about our measurements. • Basic idea... An outcome that would rarely happen if a claim were really true is good evidence that the claim is not true. • Example... I claim that 99% of adult humans are 6 feet tall or taller. • So, I’m going to take a random sample of 1000 humans and measure their heights. Then I calculate the sample mean height and get 5’ 8”. Do you agree with my claim or disagree with my claim, based on my sample statistic? • If my claim was true, it would be very rare to get most of the adult humans in a random sample of 1000 that are shorter than 6 feet. OR on the flip side, it would be very common for most of my sample to be 6 feet tall or taller.

Start with: A Pair of Hypotheses • If we flip a penny, we can agree that the probability of heads or tails is 0.50; fair. • However, some claim if we spin a penny on a table, because the heads side bulges outwards, the lack of symmetry will cause the spinning coin to land on one side more often than the other; probability is not 0.50 for each side; unfair • Some people might find this claim outrageous; completely false

Start with a research hypothesis... Null hypothesis, Ho, p = 0.50 - null hypothesis is always neutral, no change, always = - null hypothesis is always in terms of population parameter (like p or μ) Alternative hypothesis, Ha, p ≠ 0.50 - alternative hypothesis is always <, >, or ≠ - alternative hypothesis is always in terms of population parameter (like p or μ)

Like in a criminal trial... Null hypothesis, Ho, p = 0.50 - In the beginning, we assume null is true (like defendant is assumed not guilty in the beginning of a trial) until there is overwhelming evidence that suggests this is not so; then we may reject this believe if/when the evidence is clearly against it Alternative hypothesis, Ha, p ≠ 0.50

Null Hypothesis... Ho ... • The null hypothesis always gets the benefit of the doubt and is assumed to be true throughout the hypothesis-testing procedure. If we decide at the last step that the observed outcome (our sample statistic) is extremely unusual under this assumption, then and only then do we reject the null hypothesis.

Ho: p = 0.50 Ha: p ≠ 0.50 • If null hypothesis is correct, then when we spin a coin a number of times, about ½ of the outcomes should be heads. If null hypothesis is wrong, we will see either a much larger or much smaller proportion. • Let’s spin some pennies. Spin (on desk) 20 times. Count the # of heads • Calculate the sample proportion, of heads and write on board

Ho: p = 0.50 Ha: p ≠ 0.50 • Let’s look at our sampling distribution; describe using SOCS (review) • If we did this again, would be get different results? • How ‘extreme’ of a result would we need for you to not believe our null hypothesis/to reject null? • We will come back to this later in the chapter...

Practice with null and alternative hypotheses... • What’s wrong with ... • Ho: p = 0.17 Ha: p ≠ 0.19 • Ho: p = - 0.20 Ha: p < 0.15 • Ho: p > 0.45 Ha: p = 0.45 • Ho: p = 1.50 Ha: p > 1.50 • Ho: = 0.92 Ha: < 0.92

Practice: State the appropriate null hypothesis and alternative hypothesis in each case. Be sure to define your parameter each time. A recent Gallup Poll report on a national survey of 1028 teenagers revealed that 72% of teens said they rarely or never argue with their friends. You wonder whether this national result would be different in your school. So you conduct your own survey of a random sample of students at your school.

Practice: State the appropriate null hypothesis and alternative hypothesis in each case. Be sure to define your parameter each time. The proportion of people who live after suffering a stroke is 0.85. A drug manufacturer has just developed a new treatment that they claim will increase the survival rate.

Explain what is wrong in each situation and why it is wrong A change is made that should improve student satisfaction with the parking situation at COC. The null hypothesis, that there is an improvement, is tested versus the alternative, that there is no change.

Explain what is wrong in each situation and why it is wrong • A researcher tests the following null hypothesis • Ho: = 0.80

Explain what is wrong in each situation and why it is wrong A statistics instructor at COC read that 90% of all college students use social media on a regular basis. She wonders if the percent of COC students who use social media on a regular basis is different. Ho: p = 0.90 Ha: p > 0.91

Explain what is wrong in each situation and why it is wrong The Census Bureau reports that households spend an average of 31% of their total spending on housing. A homebuilders association in Cleveland believes that this average is lower in their area. They interview a sample of 40 households in the Cleveland metropolitan area to learn what percent of their spending goes toward housing. Take p to be the mean percent of spending devoted to housing among all Cleveland households. H0: p = 31% Ha: p < 31%

The Main Ingredient: Surprise • Surprise itself; when something unexpected occurs (like only making 20% of free throws when we claimed to make 90%) • Null hypothesis tells us what to expect; it’s what we believe throughout the process until we see evidence otherwise • If we see something unexpected, then we should doubt the null hypothesis • If we are really surprised, then we should rejected it altogether

Let’s go back to our penny spinning...

We have a way to measure our “surprise”... • Instead of just not surprising, kind of surprising, very surprising, etc., we have... • p-value • A p-value is a probability. Assuming the null hypothesis is true, the p-value is the probability that if the experiment were repeated many times, we would get as extreme or more extreme outcome than the one we actually got (our statistic). A small p-value suggests that a surprising outcome has occurred and discredits the null hypothesis.

P-Values A p-value is a quantitative measure of rarity of/how unlikely a finding Small p-values are evidence against Ho Large p-values fail to give evidence against Ho

P-value... all about extremes... • Understanding how to interpret a p-value is crucial to understanding hypothesis testing. • Stat Crunch will calculate the p-value, but we need to understand how the software did the calculation • The meaning of the phrase, “as extreme as or more extreme than’ depends on the alternative hypothesis

P-value... all about extremes... Three basic pairs of hypotheses...

Let’s go back to spinning coins... Ho: p = 0.50 Ha: p ≠ 0.50 • Note: the closer the number of heads is to 10, the larger the p-value • Also note the p-value for an outcome of 11 heads is the same as for 9 heads, etc.

Statistical Significance... • Most of the time, we take one more step to assess evidence against Ho • We compare the p-value to some pre-determined value (versus ‘unlikely’) called a significance level, symbol α (alpha) • Can think of this as a rejection zone (sketch)

Statistical Significance • Significance level makes ‘not likely’ more exact, more informative • Most common α levels are α = 0.05 or α = 0.01 • Interpretation: • At α = 0.05, data give evidence against Ho so strong it would happen no more than 5% of the time

Statistical Significance • If p-value is as small or smaller than α, we say data are statistically significant at level α • Note: ‘significant’ in statistics doesn’t mean important (like in English); it means not likely to happen by chance

Let’s sketch some pictures of rejection zones and p-values... • Ho: p = ... Ha: p ... • I gathered sample data, and calculated a p-value based on sample data (probability of getting that value or more extreme assuming that null hypothesis is true) • 1-sided • 2-sided

Statistically Significant Sketches • If p-value is p = 0.03... this is significant at α = 0.05 level (in rejection zone) • If p-value is p = 0.03... this is not significant at α = 0.01 level (not in rejection zone)

Interpretation/Wording Reject Ho (Null Hypothesis): This happens when sample statistic is statistically significant, p-value is too unlikely to have occurred by chance (we don’t believe null hypothesis), in the rejection zone Wording must reference all of the following for a complete interpretation... p-value, α level, reject Ho, and conclusion in context (caution about using the word ‘cause’ or ‘prove’).

Interpretation/Wording Fail to Reject Ho (Null Hypothesis): This happens when sample statistic could have occurred by chance (we do believe null hypothesis; we don’t believe the alternative), not in rejection zone Wording must reference all of the following for a complete interpretation... p-value, α level, fail to reject Ho, and conclusion in context (caution about using the word ‘cause’ or ‘prove’)

Conditions for Tests about a population proportion... • Random Sample ... randomly selected or randomly assigned • Large Sample Size; Normality (see next slide) ... npo ≥ 10 and n(1 – po) ≥ 10; the sample has at least 10 expected successes and at least 10 expected failures • Big Population (Independence) ... Population at least 10 times sample size; and each observation has no influence on any other

...if these conditions are satisfied, then we can use the Central Limit Theorem for sample proportions; distribution is ≈ Normal! That’s a great thing! • When doing a hypothesis test, you MUST check conditions... this is an essential part of the hypothesis testing process

So many p’s... Caution!

Hypothesis testing in four steps...

Work stress... According to the National Institute for Occupational Safety and Health, job stress poses a major threat to the health of workers. A national survey of restaurant employees found that 75% said that work stress had a negative impact on their personal lives. A random sample of 100 employees from a large restaurant chain finds that 68 answer “Yes” when asked, “Does work stress have a negative impact on your personal life?” Is this good reason to think that the proportion of all employees in this chain who would say “Yes” differs from the national proportion p0 = 0.75? H0: p = 0.75 Ha: p ≠ 0.75 We want to test a claim about p, the true proportion of this chain's employees who would say that work stress has a negative impact on their personal lives.

Work stress... Conditions: 1-sample proportion hypothesis test; α = 5% (rejection zone) Random Sample – stated in problem Large Sample Size/Normality - The expected number of “Yes” and “No” responses are (100)(0.75) = 75 and (100)(0.25) = 25, respectively. Both are at least 10. Big Population (Independence) - Since we are sampling without replacement, this “large chain” must have at least (10)(100) = 1000 employees.

H0: p = 0.75 Ha: p ≠ 0.75sample statistic = 68/100 Calculations for 1-sample proportion 2-sided hypothesis test; use Stat Crunch Stat, proportion stats, 1 sample, with summary z = -1.616 P-value = 0.1059

Work stress... Interpretation: Fail to reject Ho. With a p-value of 0.1059 and an α = 5%, we fail to reject the null hypothesis and conclude that there is not enough evidence to suggest that the proportion of this chain restaurant's employees who suffer from work stress is different from the national survey result, 0.75.

We want to be rich... • In a recent study, 73% of first-year college students responding to a national survey identified “being very well-off financially” as an important personal goal. A state university finds that 132 of a random sample of 200 of its first-year students say that this goal is important. • Is there evidence that the proportion of all first-year students at this university who think being very well-off is important differs from the national value, 73%? Carry out a significance test to help answer this question.

n = 200; x = 132; SRS; p = .73; = 0.66 We want to test Ho: p = 0.73 versus Ha: p ≠ 0.73 regarding the proportion of all first-year students at this university who think being very well-off is important differs from the national value of 73%.

n = 200; x = 132; SRS; p = .73; = 0.66 Conditions: 1-sample proportion hypothesis test; α = 5% (rejection zone) Random Sample/SRS – stated in problem Large Sample Size/Normality – np ≥ 10 & n (1 – p) ≥ 10 (200)(0.73) ≥ 10 & (200) (1 -0.73) ≥ 10 Big Population (Independence) – We must assume at least (10)(200) first-year students in the population.

Chapter 8: Hypothesis Testing for Population Proportions