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Coverage of Test 2 1. Inference — Confidence and Prediction Intervals

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Coverage of Test 2 1. Inference — Confidence and Prediction Intervals Coverage of test 2 1. Inference — confidence and prediction intervals (a) confidence interval for µ, known σ (b) confidence interval for µ, unknown σ (c) prediction interval for future observation, unknown σ (d) confidence interval for Binomial p 2. Inference — hypothesis testing (a) null, alternative hypotheses (b) Type I and II errors (c) one and two tailed tests (d) statistical significance versus practical importance (e) µ, known σ (f) µ, unknown σ (g) paired samples, two-sample tests (h) Binomial p (i) two-sample Binomial comparison of probabilities 3. Regression (a) purposes of regression (b) linear model assumptions (c) least squares estimates; standard error of estimate; R2 (d) hypothesis testing — t, F tests (e) confidence intervals, prediction intervals (f) checking assumptions (g) transformations — log/log, semilog models Note: You should not view the list above as exhaustive. Everything that we have discussed in class, or is in the course supplement, is fair game to appear on the test. Practice problems (1) A research organization commissioned a study that surveyed Americans and asked them if they were worried about their finances. (a) Of the 324 people surveyed whose annual incomes were between $40,000 and $120,000 (defined as middle-income), 160 reported that they were worried about their financial condition. Construct an interval estimate at the 99% level for the true proportion of Americans whose incomes are in this range that feel this way. c 2020, Jeffrey S. Simonoff 1 (b) The survey also asked 304 people whose annual incomes were less than $40,000 (defined as low-income) this question, and 210 said that they were worried about their financial condition. Is the proportion of people who feel this way for the low-income people significantly different from the proportion who feel this way for the middle-income people? Carefully state the hypotheses that you are testing and the test that you are using. (2) Air “tightness” refers to the characteristics of a home in terms of energy efficiency. While a “tight” home is obviously beneficial, it is in direct contrast to the “health” of a home, in the sense of providing an environmentally friendly, properly ventilated, home that is free from indoor pollutants. For this reason, it is of interest to try to understand the tightness of a home as a function of its characteristics, since it is difficult to measure tightness on a home-by-home basis. The following analysis is based on a sample of 66 homes in northern Louisiana (“Empirical Modeling of Air Tightness in Residential Homes in North Louisiana,” by J.J. Erinjeri, R. Nassar, M. Katz, and N.M. Witriol, Case Studies in Business, Industry and Government Statistics, 3, 37-47, 2009). The tightness of each home is measured using the CFM50, the airflow in cubic feet per minute through a blower door fan needed to create a change in building pressure of 50 pascals (normal atmospheric pressure at sea level corresponds to 101,325 pascals). The following output refers to a regression model fit with logged (base 10) CFM50 as the response variable, and the year the house was built (Year built), the living area of the house in square feet (Area), and the number of bedrooms in the house (NOB) as predictors. Regression Analysis: Logged CFM50 versus Year built, Area, NOB Analysis of Variance Source DF AdjSS AdjMS F-Value P-Value Regression 3 2.25330 0.75110 46.95 0.000 Year built 1 1.01058 1.01058 63.17 0.000 Area 1 1.05137 1.05137 65.72 0.000 NOB 1 0.01590 0.01590 0.99 0.323 Error 62 0.99179 0.01600 Total 65 3.24509 Model Summary S R-sq R-sq(adj) R-sq(pred) 0.126478 69.44% 67.96% 64.80% Coefficients Term Coef SECoef T-Value P-Value VIF Constant 16.77 1.74 9.66 0.000 c 2020, Jeffrey S. Simonoff 2 Year built -0.007001 0.000881 -7.95 0.000 1.01 Area 0.000244 0.000030 8.11 0.000 1.18 NOB 0.0260 0.0260 1.00 0.323 1.18 Regression Equation Logged CFM50 = 16.77 - 0.007001 Year built + 0.000244 Area + 0.0260 NOB Settings Variable Setting Year built 1990 Area 2000 NOB 3 Prediction Fit SEFit 95%CI 95%PI 3.40032 0.0195390 (3.36127, 3.43938) (3.14450, 3.65615) (a) Is the overall regression relationship statistically significant here? Carefully state the hypotheses that you are testing and the test that you are using. Use α = .05. (b) Do any of the individual predictors provide significant predictive power for the logged CFM50 value given the others? Carefully state the hypotheses that you are testing and the test(s) that you are using. Use α = .05. (c) What proportion of variability in logged CFM50 value is accounted for by the predictors? (d) What does the slope coefficient for the variable Year mean in terms of the rela- tionship between the year the house was built and the CFM50? Be as complete and specific as possible in answering this question exactly as it is stated. (e) A north Louisiana home buyer is considering purchasing a 2000 square foot three- bedroom home that was built in 1990. Provide her an exact interval estimate (at a 95% level) for the tightness level (in CFM50) of that house. (f) The display on the next page gives residual plots for this model fit. Are there any potential model violations that you can see in these plots? If so, what would you do to try to address them? Be as specific and complete as possible in your discussion. c 2020, Jeffrey S. Simonoff 3 (3) The following is an excerpt from an article which appeared in The Independent Mag- azine, dated November 14, 1992: A parapsychologist projected slides on to a screen in an adjacent soundproofed room. The slides were either blank rectangles or of a powerful emotionally affecting image. I had to tell through clairvoyance which type of slide — blank or affecting — was being shown on the screen in the next room. My score was 13 out of 24 ... this was enough to make me believe I had clairvoyant powers. Do you agree with the author’s conclusion? That is, does this result provide sufficient evidence to reject the hypothesis that it could have occurred simply by random chance? Carefully state the hypotheses you are testing, and the test that you are using. Use α = .05. (4) When the weather is reported for New York City we tend to think that the numbers given apply to the entire city, but that is of course not necessarily the case. Indeed, the National Weather Service has three different weather offices in New York City, at Central Park, LaGuardia Airport, and JFK Airport. Is weather the same at these different stations, or different? This can be investigated by looking at daily weather observations. The following output was obtained from Minitab related to this question. The data consist of daily average relative humidity values (expressed as a percentage of the maximum possible humidity for that temperature) for each of the 365 days between December 1, 2016 and November 30, 2017, inclusive, for JFK Airport (JFK Ave Hum) and Central Park (CP Ave Hum), respectively (the daily average relative humidity is defined to be the average of the high relative humidity and the low relative humidity for that day). c 2020, Jeffrey S. Simonoff 4 Two-Sample T-Test and CI: JFK Ave Hum, CP Ave Hum Method mu1: mean of JFK Ave Hum mu2: mean of CP Ave Hum Difference: mu1 - mu2 Equal variances are not assumed for this analysis. Descriptive Statistics Sample N Mean StDev SEMean JFK Ave Hum 365 68.0 15.2 0.79 CPAveHum 365 62.9 15.2 0.80 Estimation for Difference 95% CI for Difference Difference 5.08 (2.87, 7.29) Test Nullhypothesis H0:mu1-mu2 =0 Alternative hypothesis H1: mu1 - mu2 not = 0 T-Value DF P-Value 4.52 727 0.000 c 2020, Jeffrey S. Simonoff 5 Paired T-Test and CI: JFK Ave Hum, CP Ave Hum Descriptive Statistics Sample N Mean StDev SEMean JFK Ave Hum 365 68.019 15.163 0.794 CP Ave Hum 365 62.940 15.209 0.796 Estimation for Paired Difference 95% CI for Mean StDev SE Mean mu_difference 5.079 7.950 0.416(4.261, 5.898) mu_difference: mean of (JFK Ave Hum - CP Ave Hum) Test Null hypothesis H0: mu_difference = 0 Alternative hypothesis H1: mu_difference not = 0 T-Value P-Value 12.21 0.000 (a) Is the observed average JFK Airport average humidity significantly different from the observed average Central Park average humidity? Carefully state the hy- potheses that you are testing and the test(s) that you are using. Use α = .05. c 2020, Jeffrey S. Simonoff 6 (b) What assumptions are you making in applying your test(s)? Do they seem valid here? What might you do to address any problems you might see? (5) The 2002 paper “Comparative efficiency of insect repellents against mosquito bites,” by M.S. Franklin (New England Journal of Medicine, 347, 13–18) examined the efficacy of various insect repellents against mosquito bites. They applied repellents to the arms of 15 volunteers, which were then inserted into a cage with a fixed number of unfed mosquitoes. The elapsed time until the first bite was then recorded. One of the results of the study was that a 90% confidence interval for the expected elapsed time until the first bite when the product OFF! Deep Woods was applied was (283.8, 319.2), in minutes.
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