Stat 20: Intro to Probability and Statistics
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1 Stat 20: Intro to Probability and Statistics Lecture 19: Confidence Intervals for Percentages Tessa L. Childers-Day UC Berkeley 28 July 2014
2 By the end of this lecture... You will be able to: Estimate a 0-1 box model for a survey Explain the difference between confidence levels and probabilities Calculate confidence intervals for percentages 2 / 18
3 Recap: Box Models Up until now, we have: Had a box, whose contents were fixed (unchanging), and known Drawn tickets randomly (with or without replacement) Had a sample, whose contents were not fixed (random), and unknown Used a normal curve to calculate probabilities of having certain samples For instance: have 30,000 students at UC Berkeley, 65% of them have some student loan debt. Find the chance that 67% of those sampled have student loan debt, in a simple random sample of size 1, / 18
4 What if the box is unknown? Often, as in most surveys, we don t know the composition of the box. Then we have: A box with fixed (unchanging), and unknown contents Draw tickets randomly (with or without replacement) Have a sample, whose contents are not fixed (random) and known What can we do with this? For instance: have 30,000 students at UC Berkeley. Take a simple random sample of size 1,000 and find that 67% of those sampled have student loan debt. Still don t know box. Can you guess at it? 4 / 18
5 Example: Loan Debt 30,000 students at UC Berkeley. Take a simple random sample of size 1,000 and find that 67% of those sampled have student loan debt. Here, and in general: Draw a box model to describe the population Draw randomly, without replacement, from the box Guess that the population percentage is the sample statistic How far off is our guess? Can we know how far off it is? 5 / 18
6 Estimating the SE Can t know the SE exactly (since box, and thus SD are unknown). Can we estimate it? Used sample statistic to estimate composition of box Can we use the estimated composition to estimate the SE? Estimating the SD of the box using the sample percentages is called the bootstrap The estimate is pretty good if the sample is fairly large 6 / 18
7 Example: Loan Debt In a sample of 1,000 students, observe 67% have student loans. Estimate 67% of tickets in box are 1 s Estimate SD using bootstrap Estimate SE using estimated SD 7 / 18
8 Chance Process Recall, that for any chance process/box model Observed Value = Expected Value + Chance Error In the past: Known box Calculate EV, SE Unknown observed value Find P(observed value in a range) using normal curve A specific observed value has about a 95% chance of being between EV - 2SE and EV+2SE Density (% per unit x) Probability = 95% 95% expected E.V. 2SE E.V. + 2SE value Possible Ovserved Values 8 / 18
9 Chance Process (cont.) Recall, that for any chance process/box model Observed Value = Expected Value + Chance Error Now: Unknown box Unknown EV, SE Known observed value Find range covering expected value using normal curve The true expected value is covered by the interval from OV - 2SE to OV + 2SE with about 95% confidence Density (% per unit x) Confidence Level = 95% 95% observed O.V. 2SE O.V. + 2SE value Possible Expected Values 9 / 18
10 Chance Process (cont.) The observed value(s) are always random. The expected value is always fixed. What changes is what we know/see. Probability = 95% Confidence Level = 95% Density (% per unit x) 95% Density (% per unit x) 95% E.V. 2SE expected value E.V. + 2SE O.V. 2SE observed value O.V. + 2SE Possible Ovserved Values Possible Expected Values 10 / 18
11 Confidence Intervals In general: We can calculate a confidence interval for any population parameter We can calculate a confidence interval for any confidence level observed value ± multiple SE Multiple is z-score associated with confidence level desired Interpret: We are about % confident that the interval between lower endpt and upper endpt covers the population parameter 11 / 18
12 Example: Loan Debt Recall that we observed 67% of our sample had student loan debt. This lead to an estimated SE of 1.49%. Where is the uncertainty? Where is the randomness? Which of these causes the chance error? 12 / 18
13 Example: Loan Debt (cont.) Recall that we observed 67% of our sample had student loan debt. This lead to an estimated SE of 1.49%. We can calculate a confidence interval for the percentage of students with loan debt We can calculate a confidence interval with level 78.87% observed value ± multiple SE Multiple is z-score associated with 78.87%, or % Confidence Interval: 67% ± 1.25(1.49%) Interpret: We are about 78.87% confident that the interval between 65.14% and 68.86% covers the percentage of students with loan debt 13 / 18
14 Confidence Intervals (cont.) Think about drawing multiple samples, each of the same size. Say 200 of them. Is the observed value the same every time? How many 95% CIs can we make? observed value ± multiple SE Interpret: About 95% of our CIs (190) will cover the population parameter. About 5% (10) will not cover the population parameter Say you have an approximate 95% confidence interval from A to B. What is the probability that the population parameter is in this interval? 14 / 18
15 Be Careful! Notice the use of approximate and about : Since the SE is estimated, we know it is not correct, thus the CI is only approximate If the normal approximation doesn t work, using the z-value is incorrect, thus the CI is If the observed value is really off (by chance error) the CI will be off Also notice that we are using SRS 15 / 18
16 Examples A 1974 survey of Students at the University of Delaware found that 37% of students have brown eyes. I repeat this at UC Berkeley, a school of 30,000 students, using a simple random sample of 200 students. I find that 85 students have brown eyes. 1 The percentage of students at UC Berkeley with brown eyes is estimated as what? How much is this estimate likely to be off by? 2 If possible, find a 95% confidence interval for the percentage of all 30,000 students with brown eyes. If this is not possible, explain why. 16 / 18
17 Examples (cont.) Of the 200 students, 3 of them had green eyes. 1 The percentage of students at UC Berkeley with brown eyes is estimated as what? How much is this estimate likely to be off by? 2 If possible, find a 95% confidence interval for the percentage of all 30,000 students with brown eyes. If this is not possible, explain why. 17 / 18
18 Important Takeaways Box models can be used for surveys, where we don t know the composition of the box We use the observed data to make inferences (guesses) about the box Confidence intervals (for population parameters) are different from probabilities (for unknown draws) A confidence interval for a population parameter is given by observed value ± multiple SE Next time: What if we don t have a 0-1 box? 18 / 18
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