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## How much did you drink this weekend?

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**How much did you drink this weekend?**• 0 drinks • 1-2 drinks • 3-4 drinks • 5-6 drinks • >6**Upcoming work**• HW #9 due Sunday • Part 3 of Data Project due October 28th • Quiz #5 in class next Wednesday**Examples of One-proportion test**• Everyone (100%) believes in ghosts • More than 10% of the population believes in ghosts • Less than 2% of the population has been to jail • 90% of the population wears contacts**Examples of Two-proportion tests**• Women believe in ghosts more than men • Blacks believe in ghost more than whites • People who have been to jail believe in ghosts more than people who haven’t been to jail • Women smoke more than men • Women use facebook in the bathroom more than men**Examples of One-Sample t-test**• All Priuses have fuel economy > 50 mpg • Ford Focuses get 5 mpg on average • The average starting salary for ISU graduates >$100,000 • The average cholesterol level for a person with diabetes is 240.**Examples of two-sample t-test**• The MPG for the Prius is greater than the MPG for the Ford Focus • ISU male graduates have a greater starting salary than women • The cholesterol levels are the same for people with and without diabetes.**Margin of Error: Certainty vs. Precision**• The more confident we want to be, the larger our z* has to be • But to be more precise (i.e. have a smaller ME and interval), we need a larger sample size, n. • We can claim, with 95% confidence, that the interval contains the true population proportion. • The extent of the interval on either side of is called the margin of error (ME). • In general, confidence intervals have the form estimate± ME.**Margin of Error Problem**• It’s believed that as many as 22% of adults over 50 never graduated from high school. • We wish to see if this percentage is the same among the 25 to 30 age group. • What sample size would allow us to increase our confidence level to 95% while recuding the margn of error to only 4%.**Chapters 17**Testing Hypotheses About Proportions**ISU – Statistics 2011 Survey Results**• 55.5% of ISU students reported binge drinking in the previous two weeks • Sample size = 417 • Compared to other campuses 69.1%, believe the alcohol use at ISU is about the same. • Other campuses… • National results average about 32.2%**The Reasoning of Hypothesis Testing**• There are four basic parts to a hypothesis test: • Hypotheses • Model • Mechanics • Conclusion • Let’s look at these parts in detail…**1. Hypotheses**• The null hypothesis: To perform a hypothesis test, we must first translate our question of interest into a statement about model parameters. • In general, we have H0: parameter = hypothesized value. • The alternative hypothesis: The alternative hypothesis, HA, contains the values of the parameter we consider plausible if we reject the null. • We can only reject or fail to reject the null hypothesis. • If we reject the null hypothesis, this suggests the alternative is true.**Possible Hypotheses**• Two-tailed test • Ho: parameter = hypothesized value HA: parameter ≠ hypothesized value • One-tailed test • Ho: parameter = hypothesized value HA: parameter < hypothesized value • Ho: parameter = hypothesized value HA: parameter > hypothesized value**Is the coin in my hand a fair?**• Ho p=0.5 Ha p>0.5 • Ho p=0.5 Ha p<0.5 • Ho p=0.5 Ha p≠0.5**In the 1980s only about 14% of the population attained a**bachelor’s degree. Has the percentage changed? • Ho p=0.14Ha p>0.14 • Ho p=0.14Ha p<0.14 • Ho p=0.14Ha p≠0.14**Last year recycling rates were at 25%. The town of**Trashville claims that the new mandate, requiring everyone to recycle, has increased the recycling rate. • Ho p=0.25Ha p>0.25 • Ho p=0.25Ha p<0.25 • Ho p=0.25Ha p≠0.25**According to a census, 16% of people in the US are Hispanic.**One county supervisor believes her county has a smaller proportion of hispanics. She surveys the 493 people in her county and finds 41 are hispanic. State the hypothesis • Ho p=0.16Ha p<0.16 • Ho p=0.16Ha p≠0.16 • Ho p=0.08Ha p<0.08 • Ho p=0.08Ha p≠0.08**Testing Hypotheses**• The null hypothesis specifies a population model parameter of interest and proposes a value for that parameter. • We want to compare our data to what we would expect given that H0 is true. • We can do this by finding out how many standard deviations away from the proposed value we are. • We then ask how likely it is to get results like we did if the null hypothesis were true.**The conditions for the one-proportion z-test are the same as**for the one proportion z-interval. We test the hypothesis H0: p = p0 using the statistic where When the conditions are met and the null hypothesis is true, this statistic follows the standard Normal model, so we can use that model to obtain a P-value. One-Proportion z-Test**2. Model**• All models require assumptions, so state the assumptions and check any corresponding conditions. • Assumptions you will test • Independence • Randomization • 10% condition • Success/Failure • Determine Alpha Level**3. Mechanics**• The ultimate goal of the calculation is to obtain a P-value. • The P-value is the probability that the observed statistic value could occur if the null model were correct. • If the P-value is small enough, we’ll reject the null hypothesis. • We can define “rare event” arbitrarily by setting a threshold for our P-value. • The threshold is called an alpha level, denoted by . • If our P-value falls below that point, we’ll reject H0. • p-value < alpha level reject null • p-value > alpha level fail to reject null**According to a census, 16% of people in the US are Hispanic.**One county supervisor believes her county has a smaller proportion of hispanics. She surveys the 493 people in her county and finds 41 are hispanic.Find the p-value of your test. • p=0.0668 • p=1-0.0668 • p=-4.66 • p=0.000**P-Values**• The percentile associated with our z-value is called the p-value. • A p-value is a conditional probability • The probability of the observed statistic given that the null hypothesis is true. • The P-value is NOT the probability that the null hypothesis is true. • It’s not even the conditional probability that null hypothesis is true given the data.**According to a census, 16% of people in the US are Hispanic.**One county supervisor believes her county has a smaller proportion of hispanics. She surveys the 493 people in her county and finds 41 are hispanic. • Ho p=0.16Ha p<0.16 • Two possible conclusions: • Fail to reject null hypothesis at the 5% level. We find no evidence that suggests the local man finds water better than simply drilling • Reject the null hypothesis at the 5% level, suggesting the local man finds water better than simply drilling.**Alpha Levels**• Result: • We do not prove or disprove hypotheses. • We only suggest that the likelihood of a hypothesis being true is very very low or high. • Null is probably true. • How rare is rare? 1%, 5%, 10% chance? • Common alpha levels are 0.01, 0.05, and 0.1. • The alpha level is also called the significance level. • When we reject the null hypothesis, we say that the test is “significant at that level.”**Interpret the p-value 0.000, from our previous example of**census data • Because the p-value is so low, there is NOT sufficient evidence that the Hispanic population in this county differs from the nation • Because the p-value is so low, there is sufficient evidence that the Hispanic population in this county differs from the nation • Because the p-value is so high, there is sufficient evidence that the Hispanic population in this county differs from the nation**4. Conclusions**• We can only reject or fail to reject the null hypothesis. • If we reject the null, there is enough evidence to suggest the alternative is true, b/c the p-value is very very small. • If we fail to reject the null, there is NOT enough evidence to suggest the alternative is true, b/c the p-value is still large.**Failing to reject the null**• You should say that “The data have failed to provide sufficient evidence to reject the null hypothesis.” • Don’t say that you “accept the null hypothesis.” • In a jury trial, if we do not find the defendant guilty, we say the defendant is “not guilty”—we don’t say that the defendant is “innocent.”**HW 9 _ Problem 11**• An airline’s public relations department says the airline rarely loses passengers’ luggage. • Claim: When luggage is lost, 85% is recovered and delivered to its owner with 24 hrs. • Survey of Air Travelers: 114 of 194 people who lost their luggage on that airline were reunited with the missing items by the next day**What is the correct hypothesis test, if we want to show that**the airline’s rate is WORSE than they claim? • Ho p=.588 Ha p>.588 • Ho p=.588 Ha p<.588 • Ho p=.588 Ha p≠.588 • Ho p=.85 Ha p>.85 • Ho p=.85 Ha p<.85 • Ho p=.85 Ha p≠.85**Do the results of the survey cast doubt on the airline’s**claim of 85%? • No, because we do not reject the null hypothesis • Yes, because we reject the null hypothesis. • Yes, because we do not reject the null hypothesis • No, because we reject the null hypothesis.**Upcoming work**• HW #9 due Sunday • Part 3 of Data Project due October 28th • Quiz #5 in class next Wednesday