Chapter 20 Confidence Intervals with proportions!

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1 Chapter 20 Confidence Intervals with proportions! Statistic or Type of Variable Parameter Point Estimate Quantitative Categorical (Binary) Any Confidence Interval Point Estimate ± Margin of Error Point Estimate ± (z* or a t*) (standard error) Confidence Interval for a Mean x ± z* (σ/ n) or x ± t* (s/ n) Confidence Interval for a Proportion p ± z* (p(1-p)/n) 1

2 Confidence Interval for a Proportion Concluding Statement Based on our random sample of 25 beads, we are 99% confident that the true proportion of green beads in the bowl is between and Confidence interval: ( < p < ) or Confidence interval: ( % < p < %) Confidence Interval for a Proportion Concluding Statement Based on our random sample of beads, we are 95% confident that the true proportion of green beads in the bowl is between and Confidence interval: ( < p < ) or Confidence interval: ( % < p < %) 2

3 A recent poll of 800 college students found that 455 do not support an increase in the troops being sent to Afghanistan. Write a 90% confidence statement for this poll. Two examples... Response Variable Flight Distance Blue Eyes Type of Variable Parameter estimated Point Estimate Standard Error Margin of Error 3

4 What is the margin error for estimating the population parameter "p"? How is the margin of error affected by changes in p? How is the margin of error affected by changes in the sample size? Under what conditions can we use our techniques for creating confidence intervals to estimate "p"? 1) The sample is a SRS. 2) The number of success and the number of failures must be at least 15. Another way of saying that is: np 15 n(1 p) 15 If condition #2 fails then we can use the "plus four" method if the following conditions are met. 1) The sample is SRS. 2) The confidence level is 90% or greater. 3) The sample size is at least 10. 4

5 Plus Four Method Add four imaginary responses to the sample size. Two of these are considered "yes" or success responses, the other two are "no" or "not success" p = number of success + 2 n + 4 A random sample of 35 students were selected to compare Coke vs Pepsi. 5 students preferred Coke and 30 preferred Pepsi. Hypothesis Testing with a Proportion You start with a fair coin. What proportion of heads is assumed to land up? Suppose that you have a special rubbing technique that you think makes a coin land up heads more often than people expect. You conduct a study of 50 flips. A total of 30 land heads up. Is this result statistically signficant at the α=.10 level? H o: Words The coin is fair so the proportion of tosses landing heads is.5 Mathematically H o : p =.5 H a: The "rubbing" technique makes the coin land heads up more often. H A : p >.5 Note: 1) We are using p, the population parameter, not p hat. 2) Is this a two sided or one sided test? 5

6 Drop thumbtacks onto the table. What proportion of thumbtacks will land point up? Find a 95% confidence interval for the true proportion of tacks that will land point up. The manufacturer of these "Safety Tacks" claims that only 25% of the tacks will land face up. Conduct a hypothesis test to see if this claim is false. Words Mathematically The probability of a safety tack landing point up is.25 or 25%. H o : p =.25 H o: H a: The probability of a safety tack landing point up is not.25 or 25%. H A : p.25 Some people might make the case that we are only interested in whether the safety tacks are not as safe as the company claims. In this case, we would have a one sided test. H o: Words Mathematically The probability of a safety tack landing point up is.25 or 25%. H o : p =.25 H a: The probability of a safety tack landing point up is more than.25 or 25%. H A : p >.25 Note: We are assuming the company is telling the truth. We conduct an hypothesis test to see if that claim is false! Remember, we do not prove the company claim is true, but we might not find enough statistical evidence to say it is false. 6

7 Chapter 21 Comparing Two Proportions A recent poll of 800 college students, 450 men and 350 women, found that 230 men and 165 women support an increase in the troops being sent to Afghanistan. Find the 90% confidence statement for the difference btween men and women regarding troup increases. 1) 2) 3) 4) Press STAT Move to TESTS Menu Select option B Fill in statistics and calculate 5) Results 6) Interpret Results Based on our random sample of 450 male and 350 female college students we found no significant difference (90% confidence level) between the groups regarding their opinion about an increase in troop levels. Note: Zero (no difference) is in interval. A recent poll of 800 college students, 450 men and 350 women, found that 230 men and 165 women support an increase in the troops being sent to Afghanistan. Conduct a hypothesis test, α =.10 significance level, for the difference in opinions between men and women. H o : p m = p w H o : There is no difference between the percent of men and the percent of women favoring an increase in troops. H A : There is a difference between the percent of men and the percent of women favoring an increase in troops. H A : p m p w 1) 2) 3) 4) Press STAT Move to TESTS Menu Select option 6 Two sided test 5) Results 6) Interpret Results Based on our random sample of 450 male and 350 female college students we found no significant difference (p value >.10) between the groups regarding their opinion about an increase in troop levels. Note: A two sided hypothesis test at α =.10 should have the same conclusion as a 90% confidence interval. 7

8 Go back to the tack dropping example. Let's drop them on the floor and see if the height makes a difference in the proportion of tacks that last point side up. 8

Statistical inference provides methods for drawing conclusions about a population from sample data.

Chapter 14 Tests of Significance Statistical inference provides methods for drawing conclusions about a population from sample data. Two of the most common types of statistical inference: 1) Confidence