1. Introduction to Inference: Confidence Intervals and Hypothesis Testing Presentation 8 First Part

2. What is inference? Inference is when we use a sample to make conclusions about a population. 1. Draw a Representative SAMPLE from the POPULATION Var 1 Var 2 Va 3 459Brown28 657Red43 321Green46 213Blue47 536Blue53 2. Describe the SAMPLE 3. Use Rules of Probability and Statistics to make Conclusions about the POPULATION from the SAMPLE.

3. Population Parameters p = population proportion p = population proportion µ = population mean µ = population mean σ = population standard deviation σ = population standard deviation β 1 = population slope (we will see this in Ch. 14) β 1 = population slope (we will see this in Ch. 14) Sample Statistics = sample proportion = sample proportion = sample mean = sample mean s = sample standard deviation s = sample standard deviation b 1 = sample slope (we will see this in Ch. 14) b 1 = sample slope (we will see this in Ch. 14)

4. Two Types of Inference 1. Confidence Intervals: (Ch. 10 & 12) –Confidence Intervals give us a range in which the population parameter is likely to fall. –We use confidence intervals whenever the research question calls for an estimation of a population parameter. Example: What is the mean age of trees in the forest? Estimate the proportion of US adults who would vote for candidate A. Estimate the proportion of US adults who would vote for candidate A. 2. Hypothesis Testing: (Ch. 11 & 13) 2. Hypothesis Testing: (Ch. 11 & 13) –Hypothesis tests are tests of population parameters. Example: Is the proportion of US adult women who would vote for candidate A >50%? Example: Is the proportion of US adult women who would vote for candidate A >50%? –We can only prove that a population parameter is ‘different’ than our null value. We cannot prove that a population parameter is equal to some value. Valid Hypothesis: Is the mean age of trees in the forest > 50 years? Invalid Hypothesis: Is the mean age of trees in the forest equal to 50 years?

5. Types of CIs and Hypothesis Tests For Hypothesis Tests and C.I.’s: 1-proportion (1-categorical variable) 1-proportion (1-categorical variable) 1-mean (1-quantitative variable) 1-mean (1-quantitative variable) Difference in 2 proportions (2-categorical variables, both with 2 levels) Difference in 2 proportions (2-categorical variables, both with 2 levels) Difference in 2 means (1-quantitative and 1- categorical variable, or 2-quantitative variables, independent samples) Difference in 2 means (1-quantitative and 1- categorical variable, or 2-quantitative variables, independent samples) Regression, Slope (2-quantitative variables) Regression, Slope (2-quantitative variables) For Hypothesis Tests only: Chi-Square Test (2-categorical variables, at least one with 3 or more levels!) Chi-Square Test (2-categorical variables, at least one with 3 or more levels!)

6. Some Examples… Some Examples… Mike wants to estimate the mean high-school GPA of incoming freshman at Penn State. Mike wants to estimate the mean high-school GPA of incoming freshman at Penn State. Solution- CI for one population mean. George wants to know if the proportion of students who engage in under age drinking is greater than 25%. George wants to know if the proportion of students who engage in under age drinking is greater than 25%. Solution- Test of one proportion Ho: p ≤.25 Ha: p >.25 Doug wants to estimate the difference in the proportion of men and women who smoke. Doug wants to estimate the difference in the proportion of men and women who smoke. Solution- CI for difference in 2-proportions.

7. Interpreting CI and Hypothesis Testing Confidence Intervals: Confidence Intervals: Given the confidence level, β= 90%, 95%, 99%, etc conclude that with β % confidence the population parameter is within the confidence interval. Given the confidence level, β= 90%, 95%, 99%, etc conclude that with β % confidence the population parameter is within the confidence interval. Example: Suppose the 90% CI for age of trees in the forest is (32,45) years. Then, we are 90% confident that the true mean age of trees in the forest is between 32 and 45 years. Hypothesis Testing: Hypothesis Testing: Use the p-value to determine whether we can reject the null hypothesis. Use the p-value to determine whether we can reject the null hypothesis. We do not need to know the exact definition now, or how to calculate the p-value, but generally the p-value is a measure of how consistent the data is with the null hypothesis. A small p-value (<.05) indicates the data we obtained was UNLIKELY under the null hypothesis. We do not need to know the exact definition now, or how to calculate the p-value, but generally the p-value is a measure of how consistent the data is with the null hypothesis. A small p-value (<.05) indicates the data we obtained was UNLIKELY under the null hypothesis. Decision Rule: If the p-value is <.05 we REJECT the null hypothesis, and accept the alternative. We have a statistically significant result! If the p-value is >.05 then we say that we do NOT have enough evidence to reject the null hypothesis.

8. Second Part Confidence Intervals for 1-Proportion

9. Review of Ch.9: Sample Proportion   If np and n(1-p) are greater or equal to 10, the sampling distribution of is approximately normal with mean p and standard deviation.

10. From Sampling Distributions to Confidence Intervals… The sample proportion will fall close to the true proportion. The sample proportion will fall close to the true proportion. Thus the true proportion is likely to be close to the observed sample proportion. How close? Thus the true proportion is likely to be close to the observed sample proportion. How close? 95% of the would be expected to fall within ± 2 standard deviations of the true proportion p. 95% of the would be expected to fall within ± 2 standard deviations of the true proportion p. So if we were to construct intervals around ‘s with a width of ± 2 standard deviations these intervals would contain the TRUE population proportion 95% of the times! So if we were to construct intervals around ‘s with a width of ± 2 standard deviations these intervals would contain the TRUE population proportion 95% of the times!

11. Margin of Error & C.I. is an estimator of p but it is not exactly equal to p. is an estimator of p but it is not exactly equal to p. How far is from p? How far is from p? –Margin of Error is a measure of accuracy providing a likely upper limit for the difference between and p. –This difference is almost always less that the Margin of Error. –The almost always is translated with large probability. Usually we are talking about 90%, 95% or 99% probability. –This probability is the confidence level. For example, if the confidence level is 95%, it means that 95% of the times the difference between and p is less than the Margin of Error. (i.e. we expect 38 out of 40 samples to give a such that its difference with p is less than the Margin of Error.) Example: Based on a sample of 1000 voters, the proportion of voters who favor candidate A are 34% with a 3% Margin of Error based on a 95% confidence level. What does this tell us? Example: Based on a sample of 1000 voters, the proportion of voters who favor candidate A are 34% with a 3% Margin of Error based on a 95% confidence level. What does this tell us?

12. There is a problem here! Since p is the unknown parameter of interest, is also unknown. Thus, we substitute with the. Doing so we have that if are both ≥10, then with 95% probability we have 95% C.I. for 1-proportion (Derivation) If np and n(1-p) are ≥ 10, the sampling distribution of is approximately normal with mean p and standard deviation If np and n(1-p) are ≥ 10, the sampling distribution of is approximately normal with mean p and standard deviation From the empirical rule we have that for about 95% of the samples, is going to fall within from p, i.e. with 95% probability we have From the empirical rule we have that for about 95% of the samples, is going to fall within from p, i.e. with 95% probability we have

13. 95% Margin of Error and C.I. for p Thus, if the 95% Margin of Error is Thus, if the 95% Margin of Error is and the 95% C.I. for p is and the 95% C.I. for p is Note that we are using instead of p for the condition! Note that we are using instead of p for the condition!

14. Example 1: Obtaining a 95% C.I. for p. A sample of 1200 people is polled to determine the percentage that are in favor of candidate A. Suppose 580 say they are in favor. Construct a 95% CI for the true population proportion. A sample of 1200 people is polled to determine the percentage that are in favor of candidate A. Suppose 580 say they are in favor. Construct a 95% CI for the true population proportion. So the 95% CI for p is: So the 95% CI for p is: Conclusion: We are 95% confident that the true population proportion of those who support candidate A is between 45.5% and 51.2%. Conclusion: We are 95% confident that the true population proportion of those who support candidate A is between 45.5% and 51.2%.

15. Any C.I. for 1-proprtion Conditions: We need to have Conditions: We need to have β% CI for p : β% CI for p : –z* multiplier depends on the desired confidence level, β%. –z* is such that P(-z*<Z<z*)= β%. The most common multipliers are Interpretation: We are β% confident that the true population proportion, p, is contained within the confidence interval. Another interpretation is that for about β% samples from the population, the CI captures p. Interpretation: We are β% confident that the true population proportion, p, is contained within the confidence interval. Another interpretation is that for about β% samples from the population, the CI captures p. Margin of Error=z* times the std. error Conf. level, β%. Multiplier, z* 901.64 95 1.96 ≈ 2 982.33 992.58

16. Example 2: Obtaining a 99% C.I. for p. 300 high-risk patients received an experimental AIDS vaccine. The patients were followed for a period of 5 years and ultimately 53 came down with the virus. Assuming all patients were exposed to the virus, construct a 99% CI for the proportion of individuals protected. 300 high-risk patients received an experimental AIDS vaccine. The patients were followed for a period of 5 years and ultimately 53 came down with the virus. Assuming all patients were exposed to the virus, construct a 99% CI for the proportion of individuals protected. We have that the 99% CI for p is: We have that the 99% CI for p is: where z*= 2.58. (Can you see why using the Normal table?) where z*= 2.58. (Can you see why using the Normal table?) So the 99% CI for p =.823 ± 2.58(.0220) = (.767,.880) So the 99% CI for p =.823 ± 2.58(.0220) = (.767,.880) We are 99% confident that the true proportion of those protected by the vaccine is between 76.7% and 88.0%.

17. The Width of a Confidence Interval is affected by: n as the sample size increases the standard error of decreases and the confidence interval gets smaller. So a larger sample size gives us a more precise estimate of p. n as the sample size increases the standard error of decreases and the confidence interval gets smaller. So a larger sample size gives us a more precise estimate of p. z* as the confidence level increases (β%), the multiplier z* increases, leading to a wider CI. z* as the confidence level increases (β%), the multiplier z* increases, leading to a wider CI. So, if we want to control the length of the C.I. we can either adjust the confidence level or the sample size... So, if we want to control the length of the C.I. we can either adjust the confidence level or the sample size...

18. Question: What is an appropriate size in order to obtain a C.I. of a 95% confidence level that is not very large (i.e. with small Margin of Error)? The Margin of Error for 95% CI is equal to 2 x s.e( ). The Margin of Error for 95% CI is equal to 2 x s.e( ). Before collecting the sample, is unknown, thus we cannot calculate the exact Margin of Error. Before collecting the sample, is unknown, thus we cannot calculate the exact Margin of Error. A conservative Margin of Error is equal to A conservative Margin of Error is equal to This implies that differs from p at most ___________. This implies that differs from p at most ___________. Using the conservative Margin of Error, the length of the C.I. is equal to _____________. Using the conservative Margin of Error, the length of the C.I. is equal to _____________. How large should n be to get a 95% CI of some length L? How large should n be to get a 95% CI of some length L? n=___________. n=___________.