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Question 2 (25 Marks). Osteoporosis is a fairly common condition in post-menopausal women. The porous bones in this population are susceptible to fracture. Exercise programmes have the potential to increase the well-being of osteoporosis sufferers. A sample of 30 middle-aged osteoporotic women was randomised either to a twice-weekly physiotherapist-led exercise regime or to be a control. Physiological assessments, including a functional reach test measuring balance, were conducted on all women both before and after two months of the 'intervention' (i.e. exercise or control). a) State the appropriate null and alternate hypotheses for this trial. [2 Marks] b) Some of the output from the analysis is given on the next page. Using the relevant output, what is the difference in the mean improvement in balance between the sample of exercisers and controls? [1 Mark] c) What is the likely difference in the mean improvement in balance between the population of exercisers and controls? Interpret the relevant 95% Confidence Intervals carefully. [4 Marks] PTO 2 d) Using the relevant interval estimate and p-value, explain why there is/isn't sufficient evidence to claim that the intervention in question significantly improved balance, on average, compared to the controls. [4 Marks)
d) Using the relevant interval estimate and p-value, explain why there is/isn't sufficient evidence to claim that the intervention in question significantly improved balance, on average, compared to the controls. [4 Marks] e) What are the assumptions underlying the two sample t-test presented? [3 Marks] f) Explain why or why not the assumptions look justified based on the output provided. [2 Marks] g) What information is provided by the Tolerance Intervals provided? [4 Marks] h) Write a short paragraph summarizing the key findings of this trial. [5 Marks] Boxplot of Improvement in Balance by Regime 10- Improvement in Balance Control Exercise Regime Scatterplot of Pre and Post Balance by Regime (with line of equality) 35 830 Regime • Control Fer
Scatterplot of Pre and Post Balance by Regime (with line of equality) 35 - suelegen Regime • Control A Exercise 20 30 Post Balance PTO 3 95% Confidence Interval for difference in the population means Welch Two Sample t-test data: Improvement by Regime t = -2.092, df = 26.245, p-value = 0.04625 alternative hypothesis: true difference in means is not equal to o 95 percent confidence interval: 0.05528992 6.12688999 sample estimates:
data: Improvement by Regime t = -2.092, df = 26.245, p-value = 0.04625 alternative hypothesis: true difference in means is not equal to o 95 percent confidence interval: 0.05528992 6.12688999 sample estimates: mean in group Exercise mean in group Control 4.397375 1.306285 95% Bootstrap Confidence Interval for difference in population means osteo.boot <- osteo.df.Balance.wide %>% specify(response = Improvement, explanatory = Regime) %>% generate(reps = 1000, type = "bootstrap") %>% calculate(stat = "diff in means", order = c("Exercise", "Control")) percentile_ci <- get_ci(osteo.boot) percentile_ci ## # A tibble: 1 x 2 ## lower_ci upper_ci ## <dbl> <dbl> ## 1 0.314 6.01 95%/95% Tolerance Interval for the controls ## alpha Р x.bar 2-sided.lower 2-sided.upper ## 1 0.05 0.95 1.306285 -11.59864 14.21121 95%/95% Tolerance Interval for the Exercisers ## alpha Р x.bar 2-sided.lower 2-sided.upper ## 1 0.05 0.95 4.397375 -6.602258 15.39701