Hypothesis Tests with Proportions Chapter 10 Write down the first number that you think of for the following . . . Pick a two-digit number between 10 and 50, where both digits are ODD and the digits do not repeat. • What possible values fit this description? • Record your answer on the dotplot on the board. • What do you notice about this distribution? • Did you expect this to happen? • What proportion of the time would I expect to get the value 37 if the values were equally likely to occur? A hypothesis test will help me • Is the difference in these proportions decide! significant? How do I know if this p-hat is significantly different from the 1/8 that I expect to happen? What are hypothesis tests? Calculations that tell us if the sample These calculations (called the Is it one of the statistics (p-hat) occurs by random test statistic) willproportions tell us how sample chance or not OR . . . if it is statistically many standard deviations a that are likely to significant sample proportion is from the occur? IsStatistically it . . . population significant means that it proportion! Is it one that –isaNOT random occurrence to natural a random chancedue occurrence! isn’t likely to variation? occur? – an occurrence due to some other reason? How does murder trial tests work? Nature of ahypothesis • First begin by supposing the “effect” NOTthat present First - is assume the innocent • Next,person see ifisdata provides Then – must have sufficient evidence against the evidence to prove guilty supposition Hmmmmm … Example: murder Hypothesis tests use the same process! trial Notice the steps are the Steps: same as a confidence interval except we add 1) Assumptionshypothesis statements – which you will learn today 2) Hypothesis statements & define parameters 3) Calculations 4) Conclusion, in context Assumptions for z-test: • • YEA – These the same Have an SRS of are context assumptions as confidence Distribution is intervals!! (approximately) normal because both np > 10 and n(1-p) > 10 • Population is at least 10n Check assumptions for the •Given SRS of homes following: •Distribution is approximately normal Example A countywide water conservation because1:np=150 & n(1-p)=350 (both are campaign in a particular greater was thanconducted 10) county. month later,5000 a random •ThereAare at least homessample in the of 500 homes was selected and water usage was county. recorded for each home. The county supervisors wanted to know whether their data supported the claim that fewer than 30% of the households in the county reduced water consumption after the conservation campaign. How to write hypothesis statements • Null hypothesis – is the statement (claim) being tested; this is a statement of “no effect” or “no difference” H0: • Alternative hypothesis – is the statement that we suspect is true Ha: How to write hypotheses: Null hypothesis H0: parameter = hypothesized value Alternative hypothesis Ha: parameter > hypothesized value Ha: parameter < hypothesized value Ha: parameter = hypothesized value Example 2: (Back to the opening activity) Is the proportion of students who answered 37 higher than the expected proportion of 1/8? H0: p = 1/8 Ha: p > 1/8 Where p is the true proportion of people who answered “37” Example 3: A new flu vaccine claims to prevent a certain type of flu in 70% of the people who are vaccinated. Is this claim too high? H0: p = .7 Ha: p < .7 Where p is the true proportion of vaccinated people who do not get the flu Example 4: Many older homes have electrical systems that use fuses rather than circuit breakers. A manufacturer of 40-A fuses wants to make sure that the mean amperage at which its fuses burn out is in fact 40. If the mean amperage is lower than 40, customers will complain because the fuses require replacement too often. If the amperage is higher than 40, the manufacturer might be liable for damage to an electrical system due to fuse malfunction. State the hypotheses : H0: m = 40 Ha: m = 40 Where m is the true mean amperage of the fuses Facts to remember about hypotheses: • Hypotheses ALWAYS refer to populations (use parameters – never statistics) • The alternative hypothesis should be what you are trying to prove! • ALWAYS define your parameter in context! Activity: For each pair of hypotheses, indicate which are not legitimate & Must use parameter Must be(population) NOT equal! x explain why is a statistics (sample) a) H0 : m 15; Ha : m 15 is the population b) H0 : x 123; Ha : x 123 proportion! Must use same .1 a 1 ;asHHa 0:! –Not : isa.statistic H0 number c) P-hat parameter! d) H0 : p .4; Ha : p .6 e) H0 : pˆ .1 ; Ha : pˆ .1 Level of Significance Activity P-value - The statistic is our p-hat! • Assuming H0 is true, the probability that the statistic would have a value as extreme or more than what is actually observed Notice that this is a Why not find the probability Remember that in continuous conditional probability that the equals distributions, wep-hat cannot find a value? probabilitiescertain of a single value! P-values We can use normalcdf to • Assuming H0 find is true, the probability this probability. that the statistic would have a value as extreme or more than what is actually observed In other words . . . What is the probability of getting values more (or less) than our p-hat? pˆ pˆ Level of significance • Is the amount of evidence necessary before we begin to doubt that the null hypothesis is true • Is the probability that we will reject the null hypothesis, assuming that it is true • Denoted by a – Can be any value – Usual values: 0.1, 0.05, 0.01 – Most common is 0.05 Statistically significant – • Our statistic (p-hat) is statistically Remember that the verdict is never significant if the p-value is as small or “innocent” – so we can never decide smaller than the significance (a). that thelevel null of is true! Our “guilty” verdict. Our “not guilty” verdict. Decisions: • If p-value < a, “reject” the null hypothesis at the a level. • If p-value > a, “fail to reject” the null hypothesis at the a level. Facts about p-values: • ALWAYS make the decision about the null hypothesis! • Large p-values show support for the null hypothesis, but never that it is true! • Small p-values show support that the null is not true. • Double the p-value for two-tail (≠) tests • Never accept the null hypothesis! Never “accept” the null hypothesis! Never “accept” the null hypothesis! Never “accept” the null hypothesis! Calculating p-values • For z-test statistic (z) – – Use normalcdf(lb,ub) to find the probability of the test statistic or more extreme We will seewehow Since areto incompute the – Remember the standard normal this value tomorrow. standard normal curve, weof z’s where curve is comprised do m =not 0 need and sm, =s 1here. Draw & shade a curve & calculate the p-value: 1) right-tail test z 2) two-tail test z = 1.6 Normalcdf(1.6,∞) Double the p-value since thisP-value is a two-tailed = .0548test! z = -2.4 Normalcdf(-∞,-2.4) × 2 z P-value = .0164 At an a level of .05, would you reject or fail to reject H0 for the given p-values? a) .03 b) .15 c) .45 d) .023 Reject Fail to reject Fail to reject Reject Writing Conclusions: 1) A statement of the decision being made (reject or fail to reject H0) & why (linkage) AND 2) A statement of the results in context. (state in terms of Ha) “Since the p-value < (>) a, I reject (fail to reject) the H0. There is (is not) sufficient evidence to suggest that Ha.” Be sure to write Ha in context (words)! Example 3 revisited: A new flu vaccine claims H0: p = to .7 prevent a certain type of flu : p < .7 inHa70% ofnormalcdf(-10^99,-1.38) the people who are P-value = =.0838 Where p is the proportion of vaccinated. In true a test, vaccinated vaccinated people flu people were exposed to the the flu. HThe Since the p-value > who a, I get fail to reject 0. test for the resultstoissuggest z=Therestatistic is not sufficient evidence 1.38. Is proportion this claimoftoo high? Write that the vaccinated peoplethe who hypotheses, the 70%. p-value & do not get the calculate flu is less than write the appropriate conclusion for a = 0.05. Formula for hypothesis test: statistic - parameter Test statistic SD of parameter z pˆˆ p p 1 p n Let’s put all the steps together! Example 2 revisited: Is the proportion of people who think of the value 37 significantly higher than what we expect? Use a = 0.05. What confidence level would be equivalent to this right-tailed test with a = 0.05? Calculate this confidence interval. How do the results from the confidence interval compare to the results of the hypothesis test? Example 5: A company is willing to renew its advertising contract with a local radio station only if the station can prove that more than 20% of the residents of the city have heard the ad and recognize the company’s product. The radio station conducts a random sample of 400 people and finds that 90 have heard the ad and recognize the product. Is this sufficient evidence for the company to renew its contract? Assumptions: •Have an SRS of people •np = 400(.2) = 80 & n(1-p) = 400(.8) = 320 - Since both are greater than 10, this distribution is approximately normal. •Population of people is at least 4000. Use the parameter in the null hypothesis to check assumptions! H0: p = .2 where p is the true proportion of people who Ha: p > .2 heard the ad .225 .2 z 1.25 p value .1056 a .05 .2(.8) Use the parameter in the null hypothesis to calculate standard 400 deviation! Since the p-value > a, I fail to reject the null hypothesis. There is not sufficient evidence to suggest that the true proportion of people who heard the ad is greater than .2. The company will not renew their advertising contract with the radio station. Calculate the appropriate confidence interval for the above problem. p hat z * ( phat )(1 phat ) .225*.775 .225 1.96 n n = .225 + .041 = (.184, .266) How do the results from the confidence interval compare to the results of the hypothesis test? The confidence interval contains the parameter of .2 thus providing no evidence that more than 20% had heard the ad.