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Inference for Proportions One Sample

Inference for Proportions One Sample. Confidence Intervals. One Sample Proportions. Rate your confidence 0 - 100. Name my age within 10 years? within 5 years? within 1 year? Shooting a basketball at a wading pool, will make basket? Shooting the ball at a large trash can, will make basket?

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Inference for Proportions One Sample

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  1. Inference for Proportions One Sample

  2. Confidence Intervals One Sample Proportions

  3. Rate your confidence0 - 100 • Name my age within 10 years? • within 5 years? • within 1 year? • Shooting a basketball at a wading pool, will make basket? • Shooting the ball at a large trash can, will make basket? • Shooting the ball at a carnival, will make basket?

  4. What happens to your confidence as the interval gets smaller? The larger your confidence, the wider the interval.

  5. Point Estimate • Use a single statistic based on sample data to estimate a population parameter • Simplest approach • But not always very precise due to variation in the sampling distribution

  6. Confidence intervals • Are used to estimate the unknown population parameter • Formula: estimate + margin of error

  7. Margin of error • Shows how accurate we believe our estimate is • The smaller the margin of error, the more precise our estimate of the true parameter • Formula:

  8. Assumptions: • SRS • Normal distribution n > 10 & n(1- ) > 10 • Population is at least 10n

  9. Normal curve Formula for Confidence interval: Note: For confidence intervals, we DO NOT know p – so we MUST substitute p-hat for pin both the SD & when checking assumptions.

  10. .05 .025 .005 Critical value (z*) • Found from the confidence level • The upper z-score with probability p lying to its right under the standard normal curve Confidence levelTail AreaZ* .05 1.645 .025 1.96 .005 2.576 z*=1.645 z*=1.96 z*=2.576 90% 95% 99%

  11. Confidence level • Is the success rate of the methodused to construct the interval • Using this method, ____% of the time the intervals constructed will contain the true population parameter

  12. What does it mean to be 95% confident? • 95% chance that p is contained in the confidence interval • The probability that the interval contains p is 95% • The method used to construct the interval will produce intervals that contain p 95% of the time.

  13. A May 2000 Gallup Poll found that 38% of a random sample of 1012 adults said that they believe in ghosts. Find a 95% confidence interval for the true proportion of adults who believe in ghost.

  14. Assumptions: • Have an SRS of adults • n =1012(.38) = 384.56 & n(1- ) = 1012(.62) = 627.44 Since both are greater than 10, the distribution can be approximated by a normal curve • Population of adults is at least 10,1012. Step 1: check assumptions! Step 2: make calculations Step 3: conclusion in context We are 95% confident that the true proportion of adults who believe in ghosts is between 35% and 41%.

  15. To find sample size: However, since we have not yet taken a sample, we do not know a p-hat (or p) to use! Another Gallop Poll is taken in order to measure the proportion of adults who approve of attempts to clone humans. What sample size is necessary to be within + 0.04 of the true proportion of adults who approve of attempts to clone humans with a 95% Confidence Interval?

  16. .1(.9) = .09 .2(.8) = .16 .3(.7) = .21 .4(.6) = .24 .5(.5) = .25 By using .5 for p-hat, we are using the worst-case scenario and using the largest SD in our calculations. What p-hat (p) do you use when trying to find the sample size for a given margin of error?

  17. Another Gallop Poll is taken in order to measure the proportion of adults who approve of attempts to clone humans. What sample size is necessary to be within + 0.04 of the true proportion of adults who approve of attempts to clone humans with a 95% Confidence Interval? Use p-hat = .5 Divide by 1.96 Square both sides Round up on sample size

  18. Hypothesis TestsOne Sample Proportions

  19. Example 1: Julie and Megan wonder if head and tails are equally likely if a penny is spun. They spin pennies 40 times and get 17 heads. Should they reject the standard that pennies land heads 50% of the time? How can I tell if pennies really land heads 50% of the time? Hypothesis test will help medecide! But how do I know if this is one that I expect to happen or is it one that is unlikelyto happen? What is their sample proportion?

  20. What are hypothesis tests? Calculations that tell us if a value occurs by random chance or not – if it is statistically significant Is it . . . • a random occurrence due to variation? • a biased occurrence due to some other reason?

  21. Nature of hypothesis tests - How does a murder trial work? • First begin by supposing the “effect” is NOT present • Next, see if data provides evidence against the supposition Example: murder trial First - assume that the person is innocent Then – must have sufficient evidence to prove guilty

  22. Notice the steps are the same except we add hypothesis statements – which you will learn today Steps: • Assumptions • Hypothesis statements & define parameters • Calculations • Conclusion, in context

  23. Assumptions for z-test: Have an SRSfrom a binomial distribution Distribution is (approximately) normal YES – These are the same assumptions as confidence intervals!! Use the hypothesized parameter in the null hypothesis to check assumptions!

  24. Example 1:Julie and Megan wonder if head and tails are equally likely if a penny is spun. They spin pennies 40 times and get 17 heads. Should they reject the standard that pennies land 50% of the time? Are the assumptions met? • Binomial Random Sample • 40(.5) >10 and 40(1-.5) >10 • Infinate amount of spins > 10(40)

  25. Writing Hypothesis statements: • Null hypothesis – is the statement being tested; this is a statement of “no effect” or “no difference” • Alternative hypothesis – is the statement that we suspect is true H0: Ha:

  26. The form: Null hypothesis H0: parameter = hypothesized value Alternative hypothesis Ha: parameter = hypothesized value Ha: parameter > hypothesized value Ha: parameter < hypothesized value

  27. Example 1 Contd.:Julie and Megan wonder if head and tails are equally likely if a penny is spun. They spin pennies 40 times and get 17 heads. Should they reject the standard that pennies land 50% of the time? State the hypotheses : H0: p = .5 Ha: p ≠ .5 Where p is the true proportion of heads

  28. Example 2: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? State the hypotheses : H0: p = .2 Ha: p > .2 Where p is the true proportion that heard the ad.

  29. Formula for hypothesis test:

  30. Example 1 Contd. Test Statistics for Julie and Megan’s Data

  31. P-values - • The probability that the test statistic would have a value as extreme or morethan what is actually observed

  32. 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 α • Can be any value • Usual values: 0.1, 0.05, 0.01 • Most common is 0.05

  33. Statistically significant – • The p-value is as small or smaller than the level of significance (α) • If p > α, “fail to reject” the null hypothesis at the a level. • If p <α, “reject” the null hypothesis at the a level.

  34. Facts about p-values: • ALWAYSmake 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!

  35. Never“accept” the null hypothesis! Never“accept” the null hypothesis! Never“accept” the null hypothesis!

  36. At an αlevel of .05, would you reject or fail to reject H0 for the given p-values? • .03 • .15 • .45 • .023 Reject Fail to reject Fail to reject Reject

  37. Writing Conclusions: • A statement of the decision being made (reject or fail to reject H0) & why (linkage) • A statement of the results in context. (state in terms of Ha) AND

  38. “Since the p-value < (>) α, I reject (fail to reject) the H0. I do (do not) have statistically significant evidence to suggest that Ha.” Be sure to write Ha in context (words)!

  39. Example 1 Contd. The Decision P-Value = .342 Compare the P-Value to the Alpha Level .342 > .05 Since the P-Value is greater than the alpha level I fail to reject that spinning a penny lands heads 50% of the time. I do not have statistically significant evidence to suggest that spinning a penny is anything other than fair.

  40. What? You and Jeff Spun your pennies and got 10 heads out of 40 spins? Well that not what Meg and I got. So what now?

  41. YouDecide Joe and Jeff decide to test the same hypothesis but gather their own evidence. They spin pennies 40 times and get 10 heads. Should they reject the standard that pennies land heads 50% of the time?

  42. But we DID reject! We DID NOT reject! BOTH OF THEM!!! Who is Correct? Conclusion are based off of your data. It is important however to discuss possible ERRORS that could have been made.

  43. Errors in Hypothesis Tests Every time you make a decision there is a possibility that an error occurred.

  44. ERRORS

  45. Type I Error When you reject a null hypothesis when it is actually true. Denoted by alpha (α) -the level of significance of a test

  46. Type II Error When you fail to reject the null hypothesis when it is false Denoted by beta (β)

  47. Example 2 Revisited: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?

  48. 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 Use the parameter in the null hypothesis to calculate standard deviation! Since the p-value >α, 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.

  49. What type of error could the radio station have made? Type I Type II OR

  50. Two-Sample Proportions Inference

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