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Aerobic fitness, measured by the ability to consume oxygen, relates to the heart, blood vessels, and lungs working together to deliver oxygen-rich blood to the muscles during exercise. A high level of aerobic fitness is associated with lower risks of several diseases, including high blood pressure and coronary heart disease. In a study of aerobic fitness, a sample of 31 men involved in a physical fitness course at a university was randomly selected. The measurements included Oxygen: the oxygen intake rate (ml per kg body weight per minute),RunPulse: the heart rate while running.Our goal is to examine whether oxygen consumption is dependent upon the heart rate while running.The following is part of the SAS output:Write down the fitted regression equation. How is the estimated regression coefficient for RunPulse interpreted here? Is the intercept estimate meaningful in this context?Suppose that one subject had a runpulse of 170 and oxygen intake rate of 50 ml per kg body weight per minute, what is the residual for this subject?Construct the 95% confidence interval for the average change in oxygen intake rate when the running heart rate is increased by one. Interpret the results.Suppose that previous studies indicate that oxygen consumption decreases with an increasing heart rate while running. Does the study support this finding? Write down the hypotheses, calculate the test statistic and p-value, and state your conclusion in terms of the problem.Given that Total SS = 851.3815, SSE = 716.5368, complete the following ANOVA table:Sum of MeanSource DF Squares Square F Value P-valueModel ____ _________ ________ _______ ______Error ____ _________ ________Corrected Total ____ _________What is the estimated standard deviation of the random errors in the model?Cocaine addiction is hard to break. Addicts need cocaine to feel any pleasure, so perhaps giving them an antidepressant drug will help. A 3-year study with 72 chronic cocaine users compared an antidepressant drug called desipramine with lithium and a placebo. (Lithium is a standard drug to treat cocaine addiction and a placebo is a dummy drug). One third of the subjects, chosen at random, received each treatment. Here are the results:TreatmentCocaine relapseYesNoDesipramine1014Lithium186Placebo204If cocaine relapse is independent of which treatment a cocaine user has received, how many cocaine users taking the desipramine would be expected to experience relapse?Do cocaine addicts taking the desipramine, lithium, and placebo have different rate of relapse? Perform a hypothesis test using α = 0.01.Suppose that the investigator is interested in comparing the rate of relapse for Desipramine and placebo. Construct a 95% confidence interval for the difference in rate of relapse between Desipramine and placebo.Suppose the investigator is interested that, for the people who take Lithium, whether the probability of relapse is the same as the probability of no relapse. Perform a hypothesis test using α = 0.05.FEV (forced expiratory volume) is an index of pulmonary function that measures the volume of air expelled after one second of constant effort. The data fev.txt contains determinations of FEV on 654 children ages 3-19 who were seen in the Childhood Respiratory Disease Study in East Boston, Massachusetts. The variables in the data include:ID: subject ID numberAge: age in years Age_group:3-8 = 3-8 years, 9-12 = 9-12 years, 13+ = 13 years or aboveFEV: FEV in litersHeight: height in inchesSex:Male or FemaleSmoker: non = nonsmoker, Current = current smokerMake a boxplot of the FEV for children with age 3-8, 9-12, and 13+. Does it appear that the FEV is the same for children of different ages?Is FEV the same across the three age groups? Perform a hypothesis test to answer the question. Use α = 0.05.What assumptions are made with regard to the analysis in part b? Check whether these assumptions are violated.Is FEV more strongly related to sex or smoking status? Carry out appropriate statistical analysis to answer the question.The investigator is also interested in how height is associated with age. Construct the scatter plot of height against age. What is the relationship between height and age? Regardless of what you observed in f, fit the regression model with height as the response and age as the independent variable. What is the fitted regression equation?Test whether there is a positive correlation between age and height. Perform the hypothesis test using α = 0.05.Is it appropriate to use the above regression model? Why or why not?One of the goals of the Edinburgh Artery Study was to investigate the risk factors for peripheral arterial disease among persons 55 to 74 years of age. You wish to compare mean LDL cholesterol levels, measured in mmol/liter, among four different populations of subjects: patients with intermittent claudication or interruptions in movement, those with major asymptomatic disease, those with minor asymptomatic disease, and those with no evidence of disease at all. Samples are selected from each population with the following summary statistics:Sample sizeSample meanSample SDIntermittent claudication256.221.62Major asymptomatic disease265.811.43Minor asymptomatic disease605.771.24No disease2705.471.31Complete the following ANOVA table for the studyDFSSMSBetweenWithinTotalPerform a hypothesis test to compare the LDL cholesterol levels of the four groups at the 0.05 significance level.Short answer questions. Unless stated otherwise, each part is unrelated.In the linear regression model, it was found that the 90% confidence interval for the regression coefficient β is (1.5, 9.3). What can you say, if anything, about the p-value of test of hypotheses H0:β=0 vs.Ha:β≠0?To study the muscle mass decrease with age in women, a nutritionist randomly selected 15 women from each 10-year age group, beginning with age 40 and ending with age 79. The scatter plot of muscle mass versus age is shown below. Please raise any concern that you have for fitting a linear regression model.Medical researchers wish to study the effect of body mass on cholesterol levels in women aged 25 - 75. A random sample of 27 women in this age group was selected to participate in this study. The body mass and total cholesterol is measured for each woman and summarized in the following table. MeanStandard deviationBody mass (x)67.5 kg5.2 kgCholesterol level (y)202 mg/dl12.6 mg/dlFurthermore, it was confirmed that the relationship between body mass and cholesterol level is linear, with the estimated linear regression equation being y=84.194+bx. Calculate the estimated regression slope b, and show the complete fitted regression equation.A two-sample t test is performed with Ha: μ1>μ2 and a test statistic of t = ?1.24 is observed. Draw a sketch of the area corresponding to the p-value, and find the range of the p-value when df = 12.A teacher finds that the scores of a particularly difficult test were normally distributed with a mean of 76 and standard deviation of 14. If a score of below 60 represents a grade of F (failure), what percent of students failed the test? How many points must be added to the students’ scores so that only 5% fail?Answer T (true) or F (false).A residual plot can help to assess the validity of a linear model.A scatter plot can help to assess the validity of a linear model.If the correlation coefficient between x and y is 0.9, then linear regression is appropriate.If HA is 1 > 2 and ts = 1.6, the P-value is less than 0.5.When n = 20 and p = 0.1, I can approximate the distribution of the sample proportion with a normal distribution with mean 2 and standard deviation 1.34.When n = 36 and p = .68, I can approximate the distribution of the sample proportion with a normal distribution with mean 24.48 and standard deviation 2.8.For the same data, a 95% confidence interval for the mean is wider than a 90% confidence interval for the mean.As the sample size gets larger, the sample mean becomes closer to the population mean.As the sample size increases, the confidence interval for the mean becomes wider.The sample mean is more resistant than the median to a large change in a single data point.A p-value represents the probability that the null hypothesis is true.A p-value represents the probability that the alternative hypothesis is true.If I reject the null hypothesis, then I could have made a type I error.If I reject the null hypothesis, then I could have made a type II error.If I do not reject the null hypothesis, then I could have made a type I error.If I do not reject the null hypothesis, then I could have made a type II error.When there are only two groups in ANOVA, the F-test is equivalent to the two-sample t-test with pooled SE. ................
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