TA: Sheng Zhgang (Th 1:20) / 342 (W 1:20) / 343 (W 2:25) / 344 (W 12:05) Haoyang Fan (W 1:20) / 346 (Th 12:05) FINAL EXAM

Transcription

1 STAT 301, Fall 2011 Name Lec 4: Ismor Fischer Discussion Section: Please circle one! TA: Sheng Zhgang (Th 1:20) / 342 (W 1:20) / 343 (W 2:25) / 344 (W 12:05) Haoyang Fan (W 1:20) / 346 (Th 12:05) FINAL EXAM Problem Points Grade Total 150

2 1. Suppose that a demographic study done this year on the general population of a certain area finds that in a random sample of n = 2500 residents, X = 810 belong to a particular ethnic group. Answer each of the following. Show all work! (a) Calculate the point estimate for the true proportion π of this ethnic group in the general population, based on this sample. (2 pts) (b) Calculate the corresponding two-sided 95% confidence interval for the true proportion π of this ethnic group in the general population, based on this sample. (7 pts) (c) Interpret the meaning of this confidence interval in the context of the study. Be precise. (4 pts) (d) Further suppose that the proportion of residents belonging to this ethnic group last year was 36%. Calculate the two-sided p-value of this sample, under the null hypothesis H 0 : π = (7 pts) (e) At the α =.05 significance level, infer whether or not the null hypothesis H 0 : π = 0.36 can be rejected in favor of the alternative H A : π Clearly explain how EACH of your answers in (a) and (b) leads to this conclusion. (5 pts) (f) Interpret in context: Specifically what, if anything, has been shown about the population? Be as precise as you can. (5 pts)

3 2. The ages X of two distinct populations having an uncommon medical condition are normally distributed, i.e., X1 ~ N ( µ 1, σ 1) and X2 ~ N ( µ 2, σ 2). Investigators wish to compare their mean ages by conducting a formal test of the null hypothesis H 0: µ 1 = µ 2, versus the two-sided alternative hypothesis H A : µ 1 µ 2, at the α =.05 significance level. Random samples of individuals from both populations are independently selected; their respective ages appear below. Sample 1: {53, 62, 71} Sample 2: {28, 41, 46, 49} (a) Calculate the point estimate x1 x2 of the true difference between the two means µ 1 µ 2. (3 pts) (b) Calculate the standard error (estimate) of the sampling distribution of X1 X2. (7 pts) (c) Construct the 95% confidence interval for µ 1 µ 2, based on these sample data. (5 pts) (d) Use the appropriate distribution table to find the closest lower and upper bounds for the p-value (e.g.,.01 < p <.05). (2 pts) (e) Use BOTH (c) and (d) to infer a formal conclusion about whether or not the null hypothesis can be rejected at the α =.05 significance level. State all reasons clearly. (5 pts) (f) Interpret your conclusion in (e) in the context of the study. Exactly what (if anything) has been demonstrated? Be precise. (5 pts) (g) Suppose that, before any random sampling had been done, the investigators had legitimate medical reasons to believe that the mean age µ 1 of Population 1 is significantly older than the mean age µ 2 of Population 2. Formulate the appropriate one-sided null and alternative hypotheses, and determine the corresponding p-value for the same sample data. (3 pts) H : 0 H : A p-value =

4 3. (a) The results of a survey of a random sample of n = 120 chocolate-lovers who expressed a preference between dark, milk, and white chocolate are shown below. Dark Chocolate Milk Chocolate Conduct a Chi-squared Test to determine whether or not the null hypothesis H : π = π = π can be rejected, at the α =.05 significance level. Use the included 0 Dark Milk White table to find the closest lower and upper bounds for the p-value (Example:.01 < p <.05). (10 pts) (b) Suppose the sample data is further divided by gender, via the following 2 3 contingency table. Dark Chocolate Milk Chocolate Men Women Conduct a Chi-squared Test for the two categorical variables I = Gender (Men / Women) and J = Chocolate Preference (Dark / Milk / White) at the α =.05 significance level. Use the included table to find the closest lower and upper bounds for the p-value (Example:.01 < p <.05). (20 pts) (c) Interpret: Summarize the results of parts (a) and (b) in context. What has been shown in this formal analysis of chocolate preference overall, and its relation to gender? Be precise! (5 pts)

5 4. A study is conducted to compare the relative efficiencies of three types of alternative fuel: propane, ethanol (grain alcohol), and methanol (wood alcohol). In the study, identical automobiles in three balanced groups of nine each are given a volume of fuel equivalent to the amount of gasoline needed to drive 100 miles, and the corresponding distance traveled X (miles) on the fuel is then measured. The results are shown below. Propane Ethanol Methanol n 1 = 9 n 2 = 9 n 3 = 9 x 1 = 75.0 miles x 2 = 65.0 miles x 3 = 55.0 miles s 1 2 = 82.0 miles 2 s 2 2 = 81.0 miles 2 s 3 2 = 77.0 miles 2 Assume that the distance measurements of the k = 3 populations from which these samples were obtained are each approximately normally distributed. Furthermore, because the three sample variances s 1 2, s 2 2, and s 3 2 are fairly close in value, it is reasonable to assume equivariance of these populations, that is, σ 1 2 = σ 2 2 = σ 3 2. Given these assumptions, answer the following. (a) Using this information, complete the ANOVA table below, including the F-statistic and corresponding p-value, relative to.05 (i.e., <.05, >.05, or =.05). (15 pts) Source df SS MS F-ratio p-value Treatment Error Total Recall that, for the k groups being compared, and pooled sampled size n = n 1 + n n k, grand mean x = n 1 x 1 + n 2 x n k xk n SS Trt = n 1 ( x 1 x ) 2 + n 2 ( x 2 x ) n k ( x k x ) 2, df Trt = k 1 SS Err = (n 1 1) s (n 2 1) s (n k 1) s k 2, df Err = n k (b) Test the null hypothesis H 0 : µ 1 = µ 2 = µ 3 at the α =.05 significance level. Interpret in context: Exactly what conclusion can be inferred in this comparison of the three fuels? (5 pts)

6 5. As part of a small study analyzing potential associations between different risk factors for cardiovascular disease, a physician measures the variables X = amount overweight (lbs) and Y = serum cholesterol level (mg/dl) on a random sample of n = 5 high-risk patients, and organizes the data in the following table with corresponding summary statistics. X x = 20 s x 2 = 250 Y y = 240 s y 2 = 1650

7 (a) Sketch and label a scatterplot of the data points on the set of axes above. (5 pts) (b) Compute the sample covariance s xy. Show all work. (4 pts) (c) Compute the sample correlation coefficient r. Use it to determine whether or not X and Y are linearly correlated; if so, classify as positive or negative, and as weak, moderate, or strong. (4 pts) (d) Determine the equation Yˆ = ˆ β ˆ 0 + β1 X of the least squares regression line for these data. Calculate the fitted response values y ˆi, and sketch a graph of this line on the same scatterplot above. Show all work. (12 pts) (e) Calculate the residuals ˆ 2 ei = yi yi, and the residual sum of squares SS ˆ Error = ( yi yi). Show all work. How does this value compare with SS Error for any other line that estimates the data? Be as precise as possible. (6 pts) n i= 1 (f) Calculate the sample coefficient of determination r 2, and interpret its value in the context of evaluating the fit of this linear model to the sample data. Be as precise as possible. (4 pts)

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10 Chi-squared scores corresponding to selected right-tailed probabilities of the χ distribution 2 df Righttailed area 0 χ 2 -score df

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12 One Sample POPULATION PARAMETER Null Hypothesis H 0 : θ = θ 0 SAMPLE STATISTIC Point Estimate ˆ θ = f(x 1,, x n ) CRITICAL VALUE (2-sided) 1 MARGIN OF ERROR = product of these two factors: STANDARD ERROR (estimate) 2 Mean* H 0 : μ = μ 0 ˆμ = x = x i n Proportion H 0 : π = π 0 ˆ π = p = X n, where X = # Successes n 30: t n 1, α /2 or z α /2 n < 30: t n 1, α /2 only n 30: z α /2 ~ N(0, 1) n < 30: Use X ~ Bin(n,π). (not explicitly covered) Any n: s / n n 30: For Confidence Interval: ˆ π (1 ˆ π) n For Acceptance Region, p-value: π (1 π ) n 0 0 Two Independent Samples Two Paired Samples 3 Null Hypothesis H 0 : θ 1 θ 2 = 0 Point Estimate ˆ θ ˆ θ 1 2 Means* H 0 : μ 1 μ 2 = 0 x1 x2 CRITICAL VALUE (2-sided) 1 STANDARD ERROR (estimate) 2 n 1, n 2 30: tn1+ n2 2, α / 2 or z α /2 n 1, n 2 30: s 2 1 / n 1 + s 2 2 / n 2 n 1, n 2 < 30: Is σ = σ 2? n 1, n 2 < 30: Informal: 1/4 < s 2 1 /s 2 2 < 4? 2 s pooled 1 / n / n 2 Yes t n1+ n2 2, α / 2 No Satterwaithe s Test n 1, n 2 30: z α /2 Proportions H 0 : π 1 π 2 = 0 ˆ π ˆ 1 π 2 n 1, n 2 < 30: (or use Chi-squared Test) Fisher s Exact Test (not explicitly covered) where s pooled 2 = (n 1 1) s (n 2 1) s 2 2 n 1 + n 2 2 n 1, n 2 30: For Confidence Interval: ˆ π1(1 ˆ π ˆ ˆ 1) n1 + π2(1 π2) n2 For Acceptance Region, p-value: ˆ π pooled (1 ˆ π ) 1 n + 1 n pooled 1 2 where ˆ π = (X 1 + X 2 ) / (n 1 + n 2 ) pooled k samples (k 2) Null Hypothesis H 0 : θ 1 = θ 2 = = θ k Independent Dependent (not covered) Means H 0 : μ 1 = μ 2 = = μ k F-test (ANOVA) Repeated Measures, Blocks Proportions H 0 : π 1 = π 2 = = π k Chi-squared Test Other techniques 1 For 1-sided hypothesis tests, replace α /2 by α. 2 For means, always use the actual standard error if known either σ / n or σ 2 2 / n σ / n with the Z-distribution. 3 For Paired Means: Apply the appropriate one sample test to the pairwise differences D = X Y. For Paired Proportions: Apply McNemar s Test, a matched version of the 2 2 Chi-squared Test. * If normality is not established, then use a transformation, or a nonparametric Wilcoxon Test on the median(s).