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Lecture 13

COMM 291 Lecture 13: Lectures_13_14_15

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COMM 291
Jonathan Berkowitz

COMMERCE 291 – Lecture Notes 2014 – © Jonathan Berkowitz Not to be copied, used, or revised without explicit written permission from the copyright owner. Summary of Lectures 13, 14, and 15 Setting the stage for statistical inference: We previously defined (in Chapter 3) the terms population, sample, parameter and statistic/estimate. Now we begin our study of statistical inference. Inference means “generalization”, from the sample to the population. It is based on answering the questions, “How trustworthy are the results? Would the results persist if the study were repeated many times?” A parameter has a value that doesn’t change. A statistic (i.e. estimate) will give a different value each time a new sample is taken – this is called sampling variability. The distribution of values that a statistic can take based on all possible samples of the same size from the same population is called the sampling distribution. We will discuss this shortly. What makes an estimate a “good” estimate? Two properties: unbiasedness and low variability. Unbiased is related to the idea of accuracy. If you sample over and over again, does the average of the estimates get closer to the true population parameter? For example, if you get on the bathroom scale six times and take the average of the six readings to be your weight, you expect to be close to your true weight, assuming that the scale is properly calibrated (i.e. zeroed) – in other words, the scale is unbiased. If the scale is set to read –2 kg with no weight on it, the scale is biased. Low variability is related to the idea of precision. Are all the estimates from the various samples are close to one another? In the previous example, precision means that the six readings on the bathroom scale are nearly the same (regardless of whether that value is your true weight!). To reduce bias, use random sampling. To reduce variability, use a larger sample. 1 Chapter 11. Confidence Intervals For Proportions A Main Objective of Inference: Draw conclusions about a larger universe based on the available data. How do we do this? Based on sample information, compute a "statistic" that can be used as a good guess (i.e. "estimate") of the unknown parameter in a larger population of interest. From now on we will use the word "estimate" to refer to such a "statistic". Notation for some common parameters and their estimates Parameter Estimate µ: population mean 𝑥̅: sample mean σ: population standard deviation s: sample standard deviation (SD) p : population proportion 𝑝̂: sample proportion µ – µ 𝑥̅ − 𝑥̅ 1 2 1 2 p1– p 2 𝑝1 − 𝑝2 Confidence Intervals Point estimates such as 𝑥̅ and 𝑝̂, are based on samples, and hence have sampling variability. That is, if you sampled again, you would get different data and therefore different point estimates. Confidence intervals are an excellent way of quantifying this sampling variability and expressing it as a margin of error. The probability that a point estimate equals the true parameter value is actually zero, but the probability that an interval estimate “captures” the true parameter value can be made as high as you like, say, for example, 95%. For example, if a physician gives a pregnant woman her “due date” as a single day, the chance of a correct prediction is very small. But if, instead, the woman is given two-week range as a due date, the chance of a correct prediction is very high. Every opinion poll uses this strategy. There is always a disclaimer of the form, “the poll is estimated to be accurate to within 3 percentage points 19 times out of 20.” This is actually a 95% confidence interval. The pollster is quite sure (95% in fact) that his/her poll will not miss the true percentage by more than 3% – that’s the poll’s margin of error. The margin of error reminds you that effects may not be as large as the point estimate indicates; they could be much smaller, or they could be much larger. But they remind you not to get too excited about the accuracy of a point estimate. 2 So now that you know why you should compute confidence intervals, we’re ready to discuss how to compute them. We begin with a confidence interval for the population proportion p. (In Chapter 13, we will construct a confidence interval, for the population mean . In later Chapters we will construct confidence intervals for the difference between two proportions or two means. The basic structures are the same for all the cases we will encounter so learn it well here!) The parameter of interest is p, and it is unknown. It represents the true proportion of successes in the population and we will need to estimate it from the data, using the sample proportion of successes, 𝑝̂. How far away from p do we think 𝑝̂ is? Use the sampling distribution, an𝑆𝐷 𝑝̂ = √ 𝑝𝑞and the 68-95-99.7 Rule 𝑛 Thus in repeated samples: About 68% of 𝑝̂'s will be within one SD(𝑝̂) of p. About 95% of 𝑝̂'s will be within two SD(𝑝̂)'s of p. Since we don't know p and q in the SD formula, use SE instead. Thus any 𝑝̂ has a 95% chance of being within 2SEs of p. 𝑝𝑞 Or, 𝑝̂ ± 2 has a 95% chance of capturing p. 𝑛 This is called a Confidence Interval for p. There are many confidence intervals (C.I.), one for each type of parameter or estimation situation. All our confidence intervals will have the same form: Point Estimate  Margin of Error Example: During a class, JB picks up his coffee cup 50 times; 20 times he actually drinks from it. Compute a 95% CI for the true proportion of times JB drinks from the cup. 𝑝̂ = 20/50 = 0.40 0.40 (0.60) 0.40 ± 2√ 50 = 0.40 ± 0.14 or (0.26,0.54) or (26%,54%) 3 What if we want a higher level of confidence? Change the "multiplier", which the textbook call the "critical value" and denotes by z*. In practice, only three choices are ever used: CI z* 90% 1.645 95% 1.96 99% 2.576 (You can find these values, approximately, in the Z-table; but it is easier just to memorize the three values, especially the middle one!) Summary: "One-proportion z-interval (more commonly known as "one-sample confidence interval for a proportion) is: 𝑝̂ ± z*𝑞 , where z* is chosen based on 𝑛 the desired "level of confidence" (i.e. 90%, 95%, 99%). Two Notes on Interpretation: A Confidence Interval is a statement about the estimate and the sample it came from, not about the parameter. The 95% CI for p is an interval has a 95% chance of containing p. This is not the same as saying that there is a 95% chance that p is in the interval. The probability has to do with your ability, using your random sample, to correctly capture the parameter p. The parameter does not vary, but the confidence interval does. A Confidence Intervals contains a range of believable, sensible or plausible values of the true, but unknown, proportion p. It is an interval estimate that has a high likelihood of containing the true population parameter. Assumptions and Conditions As always, the sample must be a RANDOM sample. Sample size: • 10% Condition: n should be no more than 10% of the population • Success/Failure Condition: 𝑝̂> 10 and n𝑞̂> 10. 4 Computing sample size for a required margin of error: 𝑝𝑞 Denote the margin of error (or “plus or minus number”) by ME: ME = z*√ 𝑛 ∗ 2 Solving for n gives𝑛 = (𝑧 ) 𝑝𝑞̂ 𝑀𝐸 2 If you have a value o𝑝̂from previous work, substitute it in here, along with the desired z* and ME and compute n. If𝑝̂is unknown, use the most conservative value (i.e. the value that will give you the largest n); that value𝑝̂= 0.5. (𝑧∗) (0.5 (0.5) Then 𝑛 = 2 𝑀𝐸 For 95% confidence (i.e. “19 times out of 20”)* z = 1.96, or approximately 2. Substituting 2 into the above equation gives a handy Rule of Thumb, to be used as a guideline for sample size determination. 𝑛 ≈ 1 or 𝑀𝐸 ≈ 1 (for 95% confidence) 𝑀𝐸 2 √𝑛 Remember that this gives approximate values only. The following gives the approximate required sample sizes for various margins of error. For E = ±10%, need n ≈ 100 For E = ± 5%, need n ≈ 400 For E = ± 3%, need n ≈ 1,100 (I just remember it as about 1,000) For E = ± 2%, need n ≈ 2,500 The small improvement in precision for 3% to 2% requires more than doubling the sample size from 1,100 to 2,500! This explains why most survey samples are 1000 in size. 5 Chapter 12. Testing Hypotheses About Proportions Hypothesis Tests (also called Tests of Significance) address the issue of which of two claims or hypotheses about a parameter (or parameters) is better supported by the data. For example, is a target mean being achieved or not; does a treatment group have a higher mean outcome than a control group; is there a greater proportion of successes in one group than another? All tests have the following components: 1. Hypotheses: Ho: null hypothesis Ha: alternative hypothesis These are statements about population parameters. The null hypothesis represents “no change from the current position” or the default position. The alternative hypothesis is the research hypothesis. The burden of proof is on the investigator to convince the skeptic to abandon the null hypothesis. H o specifies a parameter and suggests a value for it. H a gives values of the parameter that would be believable if we rejected the null hypothesis Example: H o p = p o H a p ≠ p o 2. Test Statistic: Uses estimate(s) of the parameter(s), the standard error(s) of the estimate(s), and information in the null hypothesis and puts them together in a “neat package” with a known distribution. Certain values of this test statistic will support H o while others will support H .a 3. P-value: a probability that judges whether the data (via the value of the test statistic) are more consistent with H or oith H . a Note: The P-value assumes that H is truo and then evaluates the probability of getting a value of the test statistic as extreme as or more extreme than what you observed. The P-value is not the probability that H io true. The smaller the P-value, the greater the evidence against the null hypothesis. 4. Conclusion: a statement of decision in terms of the original question. It is not enough simply to write “reject H o 6 Let's develop our first hypothesis test: Examples: Is a coin "fair"? Are daily changes in the Dow Jones Industrial Average (DJIA) equally likely to be up as down? For both of these situations the hypotheses are: Ho: p = 0.5 Ha: p ≠ 0.5 Compute 𝑝̂based on a random sample of size n. Compute SD( 𝑝̂) and use it to see how far away from 0.5 (i.e. the o value) your𝑝̂ is. This will tell you if 𝑝̂is an unlikely or surprising value. Example: DJIA continued. From 1112 trading days, the DJIA is up 573 days. So 𝑝̂ = 573/1112 = 0.5153 or 51.53%. SD( 𝑝) = √ 𝑝0 0 = √ (0.5 (0.5)= 0.015 𝑛 1112 So 𝑝̂ is about 1 SD(𝑝̂) away from po. Let's be more accurate: 𝑝−𝑝 0 = 𝑝−𝑝 0 = 0.5153−0.5 = 1.02 𝑆𝐷(𝑝) 𝑝0 0 0.015 √ 𝑛 How "likely" a value is 1.02? Think back to Sampling Distributions (Chapter 9); 𝑝−𝑝 To standardize 𝑝, we use 𝑍 = 𝑝𝑞 √ 𝑛 Here p and q=1-p are the unknown, since p is the proportion in the population. But if o is a reasonable choice p and q=1-p, then 𝑝−𝑝 0 should behave like a Z. 𝑝 0 0 √ 𝑛 What does it mean to “behave like a Z?” It means that the value should be something likely to be found on a Z-table, i.e. a value between about -2 and 2. Use Table Z to find the area to the right of 1.02; that area is found to be 0.1539. Since if the DJIA is not random it might be up more than half the time or down more than half the time, we need the area to the left of -1.02; that is, the flip-side of the Z curve. So the total area is double the single tail area, giving 2x0.1539 = 0.3078 or about 31%. This probability is called the . P-value: how "surprising" are the data we observed if, in fact, the null hypothesis were true. A low P-value means the data are unlikely given the H volue so we reject H . o A high P-value means the data support H soowe do not reject H . o How low is "low"? How "unlikely" is "unlikely"? What is "reasonable doubt"? 7 1 chance in 20? 1 chance in 100? 1 chance in 1000? 8 One-sample z-test for a single population proportion, p (Note: The text calls this the one-proportion z-test) H o p = po H a p ≠ po 𝑝𝑞̂ Remember that the confidence interval for p is: 𝑝̂ ± z* √ 𝑛 𝑝−𝑝 0 Test statistic: Z = (This test statistic has a z-distribution.) √ 𝑝0 0 𝑛 Note the difference between the test statistic and the confidence interval. In the test 𝑝𝑞̂ 𝑝0 0 statistic, the SE of 𝑝̂ is used ( ), but hypothesis test uses √ . That’s because in a 𝑛 𝑛 hypothesis test we compute the test statistic, assuming H is true. o Substitute in the values and compute the test statistic; call it z-stat. P-value = 2 x Pr (z >| z-stat|) If P-value is "small", then reject o . Alpha Level and Significance The threshold for a P-value (to decide whether it is "small") is called the alpha level (α) or the level of significance. Sometimes it is stated as a percentage rather than a proportion. Usually α = 0.05, but 0.10 or 0.01 are also used. If P-value < α, then reject o ; i.e. the result is "statistically significant" If P-value > α, then do not reject Ho Alternative Hypothesis Our H a p ≠ p os called a two-sided or two-tailed alternative. One-sided or one-tailed alternatives are also possible; H :ap > p o or Ha: p < po. You need to pick one of these three alternatives at the beginning of the test. The choice depends on the wording of the question. For either Ha: p > poor H a p < po , you compute the same test statistic; but the P-value = Pr (z > |z-stat|), which is exactly half the P-value of the two-sided a : p ≠op . That is, the P-value for a two-sided alternative two times the P-value for a one-sided alternative. Isn't that easy? As with the confidence interval, this test assumes large samples, and works well when np 0 10 and nq > 00.
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