STAT 113: EXAM 2 PRACTICE PROBLEMS SOLUTION Inference Foundations. - PDF document

STAT 113: EXAM 2 PRACTICE PROBLEMS SOLUTION Inference Foundations. Parameters and Statistics. State whether the quantity described is a param- eter or statistic and give the correct notation. (These are exercises 3.1-3.5 from the text) (1) Average household income for all houses in the US, using data from the US Census (2) Correlation between height and weight for players on the 2010 Brazil World Cup team, using data from all 23 players on the roster (3) Proportion of people who use an electric toothbrush, using data from a sample of 300 adults (4) Proportion of registered voters in a county who voted in the last election, using data from the county voting records (5) Average number of television sets per household in North Carolina, using data from a sample of 1000 households. Sampling Distributions. (6) (Exercise B.10 from the text) The GRE (Graduate Record Exam) is like the SAT exam except it is used for application to graduate school instead of college. The mean GRE scores for all examinees tested between July 1, 2006, and June 20, 2009, are as follows: Verbal 456, Quantitative 590, Analytic Writing 3.8. If we consider the popula- tion to be all people who took the test during this time period, are these parameters or statistics? What notation would be appropriate for each of them? Suppose we take 1000 different random samples, each of size n = 50, from each of the three exam types and record the mean score for each sample. Where would the distribution of sample means be centered for each type of exam? Date : November 18, 2015. 1

2 SOLUTION These are parameters, since they are computed for all examinees in the time period in question, which is defined to be the population. We could write, for example, µ V = 456 µ Q = 590 µ AW = 3 . 8 The distribution of sample means for verbal would be centered at the population mean for verbal (456). Likewise for the other tests. Figure 1 contains sample proportions of one value of a binary response vari- able based on many random samples of size n = 35 from a population. The next six questions refer to this figure. Figure 1. Sample Proportions from Samples of Size n = 35 (7) What does one dot on the sampling distribution represent? Each dot comes from a sample of size 35 and represents the sample pro- portion for that sample. (8) Estimate the population proportion from the dotplot. Since sample proportions from many random samples center around the popula- tion parameter, we can infer that the population parameter is near the center of the distribution of sample proportions. That is, it is roughly 0.63 to 0.65 or so. (9) Estimate the standard error of the proportions. The standard error is the standard deviation of the distribution of sample proportions. Visually inspecting the distribution, the points that are about one standard deviation from the mean in either direction appear to be at roughly 0.56 and 0.72 or so, which represents a span of two standard deviations. One standard deviation is therefore about 0.08 or so.

STAT 113: EXAM 2 PRACTICE PROBLEMS 3 (10) For each of the following sample proportions, indicate whether it is (a) Reasonably likely to occur for a sample of this size, (b) Unusual but might occur occasionally, or (c) Extremely unlikely to occur (i) ˆ p = 0 . 45 This is an unusual value to get, though it seems to occur occasionally. (ii) ˆ p = 0 . 98 There are no values this extreme in the sample pro- portions in the plot. It is a full standard deviation past the largest value observed, so it is extremely unlikely to occur. (iii) ˆ p = 0 . 65 This is right near the center of the distribution and occurs relatively frequently. (11) If samples of size n = 70 had been used instead of n = 35, which of the following would be true? (a) The sample statistics would be centered at a larger proportion. (b) The sample statistics would be centered at roughly the same proportion. (c) The sample statistics would be centered at a smaller proportion. The sample size does not affect the center of the sampling distribu- tion, so the sample statistics would still be centered at roughly the same value. (12) If samples of size n = 70 had been used instead of n = 35, which of the following would be true? (a) The sample statistics would have more variability. (b) The variability in the sample statistics would be about the same. (c) The sample statistics would have less variability. As the sample size goes up, the variability in the sample statistics goes down. So doubling the sample size would result in less variabil- ity in the sample statistics. Confidence Intervals. (13) (B.11) A recent national telephone survey reports that 57% of those surveyed think violent movies lead to more violence in society. The survey included a random sample of 1000 American adults and re- ports: “The margin of sampling error is ± 3 percentage points with a 95% level of confidence.” (i) Define the relevant population and parameter. Based on the data given, what is the best point estimate for this parameter.

4 SOLUTION (ii) Find and interpret a 95% confidence interval for the parameter defined in (i). (14) (B.16) In a random sample of 450,000 U.S. adults the proportion of people who say they exercised at some point in the past 30 days is ˆ p = 0 . 726 with a standard error of 0.0007. Find and interpret a 95% CI for the proportion of U.S. adults who have exercised in the last 30 days. A 95% confidence interval is given by ˆ p ± 2 · SE = 0 . 726 ± 2(0 . 0007) = 0 . 726 ± 0 . 0014, which gives an interval from 0.7246 to 0.7274. We are 95% confident that the proportion of all US adults who have exercised at some point in the last 30 days is between 0.7246 and 0.7274. The confidence interval is very narrow because the sample size (over 450,000) is so large. (15) (modified from 3.65) Identify whether each of the following samples is a valid bootstrap sample from this original sample: 17,10,15,21,13,18. If it could not be obtained, explain why not. (i) 10, 12, 17, 18, 20, 21 (ii) 10, 15, 17 (iii) 18, 13, 21, 17, 15, 13, 10 (iv) 15, 10, 21, 24, 15, 10 (v) 13, 10, 21, 10, 18, 17 Bootstrap samples are the same size as the original sample, and so (ii) and (iii) are not valid. Samples (i) and (iv) are not valid as they contain one or more values not in the original sample (boot- strap samples are drawn with replacement from the original sample). Sample (v) is valid. (16) (modified from 3.69) Figure 2 represents a bootstrap distribution of sample correlations. Give a point estimate for the population correlation, and estimate a 95% confidence interval two ways: (i) by first estimating the standard error, and (ii) directly from the appropriate quantiles of the bootstrap distribution. (17) (B.29) Given a specific sample to estimate a specific parameter from a population, what are the expected similarities and differences in the corresponding sampling distribution (using the given sample size) and bootstrap distribution (using the given sample)? In partic- ular, for each aspect of a distribution listed below, indicate whether the values for the two distributions (sampling distribution and boot- strap distribution) are expected to be approximately the same or different. If they are different, explain how. (i) The shape of the distribution

STAT 113: EXAM 2 PRACTICE PROBLEMS 5 Figure 2. A Bootstrap Distribution of Sample Correlations (ii) The center of the distribution (iii) The spread of the distribution (iv) What one value (or dot) in the distribution represents (v) The information needed in order to create the distribution Hypothesis Testing. (17) (modified from 4.21-4.25) The ICUAdmissions dataset contains in- formation about a sample of patients admitted to a hospital Intensive Care Unit (ICU). For each of the research questions below, define any relevant parameters and state the appropriate null and alternative hypotheses. (i) Is there evidence that mean heart rate is higher in male ICU patients than in female ICU patients? (ii) Is there a difference in the proportion who receive CPR based on whether the patients race is white or black? (iii) Is there a positive linear association between systolic blood pressure and heart rate? (iv) Is either gender over-represented in patients to the ICU or is the gender breakdown about equal? (v) Is the average age of ICU patients at this hospital greater than 50? (18) (modified from B.7) How much of an effect does your roommate have on your grades? In particular, does it matter whether your roommate brings a videogame to college? A study examining this