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How much did you drink this weekend?

How much did you drink this weekend?

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How much did you drink this weekend?

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  1. How much did you drink this weekend? • 0 drinks • 1-2 drinks • 3-4 drinks • 5-6 drinks • >6

  2. Upcoming work • HW #9 due Sunday • Part 3 of Data Project due October 28th • Quiz #5 in class next Wednesday

  3. 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

  4. 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

  5. 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.

  6. 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.

  7. 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.

  8. 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%.

  9. Chapters 17 Testing Hypotheses About Proportions

  10. 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%

  11. 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…

  12. 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.

  13. 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

  14. 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

  15. 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

  16. 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

  17. 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

  18. 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.

  19. 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

  20. 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

  21. 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

  22. 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

  23. 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.

  24. 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.

  25. 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.”

  26. 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

  27. 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.

  28. 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.”

  29. 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

  30. 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

  31. 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.

  32. Upcoming work • HW #9 due Sunday • Part 3 of Data Project due October 28th • Quiz #5 in class next Wednesday