Page 591 - Probability and Statistical Inference

568 12. Large-Sample Inference

12.4.3 Suppose that X , ..., X are iid with the common N(θ, θ ) distribu-
2
n
1
+
tion where the parameter θ ∈ ℜ . Let us denote Derive the
asymptotic (as n → ∞) distribution Find the variance stabiliz-
ing transformation of T .
n
12.4.4 Verify the result given by (12.4.18).
12.4.5 A researcher wanted to study the strength of the correlation coef-
ficient (ρ) between the proficiency in the two specific courses, namely first-
year physics (X ) and calculus (X ), in a college campus. From the large pool
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2
of first-year students enrolled in these two courses, thirty eight students were
randomly picked and their midterm grades in the two courses were recorded.
From this data, we obtained the sample correlation coefficient r = .663. As-
sume that (X , X ) has a bivariate normal distribution in the population. Test
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2
whether α can be assumed to exceed 0.5 with an approximate level a = .10.
12.4.6 The data on systolic blood pressure ( X ) and age ( X ) for a sample
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2
of 45 men of similar health conditions was collected. From this data, we
obtained the sample correlation coefficient r = .385. Obtain an approximate
95% confidence interval for the population correlation coefficient ρ. Assume
bivariate normality of (X , X ) in the population.
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1
12.4.7 The strength of the right and left grips, denoted by X and X re-
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1
spectively, were tested for 120 auto accident victims during routine therapeu-
tic exams in a rehab center. The observed values of X and X were both
1 2
coded between zero and ten, a low (high) value indicating significant weak-
ness (strength) in the grip. Assume bivariate normality of (X , X ) in the popu-
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1
lation. The data gave rise to the sample correlation coefficient r = .605. Test
whether ρ can be assumed 0.5 with an approximate level α = .10.