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MATH102 11SP Midterm ch710
A [ answers in web view ]
Total points: 70
Let the words of my mouth and the meditation of my heart
Be acceptable in Your sight, O LORD, my Rock and my Redeemer.
 Psalm 19:14
 Please show all your work! No partial credit will be given for
incorrect answers with no work shown.
 Please draw a box around your final answer.
 You are only permitted to use your own calculator and writing implements.
Cell phones should be muted and left in your pocket or bag.
 All relevant tables are attached to the back. You may detach them for your
reference.
 Assume α = 0.05 everywhere unless indicated otherwise.
 For ttests on two groups, if the df is not given, you may use the
conservative estimate of df = min(n_{1}, n_{2})  1.

Does HBE concentration in men increase after they exercise?
Data from a study of 6 men are below.
 Mean  SD 
Before: 
42  58  38 
50  49  48 
47.5  6.921 
After: 
47  57  44 
53  49  53 
50.5  4.722 
 State the null and alternative hypotheses, both in words and in
notation. [3]
H_{0}: μ_{d} ≤ 0
(presuming d = after  before; you could also subtract in the other order),
HBE concentration does not increase after exercise.
H_{A}: μ_{d} > 0,
HBE concentration does increase after exercise.
 What statistical test is be appropriate to test the hypothesis?
Should it be 1tailed or 2tailed? [2]
Dependent ttest on pairwise differences. 1tailed.
 Run the test (either pvalue or classical approach) and
draw a conclusion. [4]
mean diff = 3.00, SD of diffs = 2.898.
SE_{d} = 1.183, t = 2.535, df = 5, onetailed.
Pvalue approach: ⇒ 0.025 < p < 0.05 (real p = 0.026).
Classical approach: t(0.05) = 2.02
 Interpret your conclusion in the context of the original
research question.
Please use complete English sentences. [2]
Reject H_{0},
HBE concentration rises after exercise.
 What assumptions did you rely upon in conducting the test?
[2]
Random sampling of men; change in HBE concentration is
normally distributed.

A factory needs to ensure that the widgets it produces have variance no more
than 2.5mm^{2}. An inspector from corporate headquarters randomly
selects 41 widgets from the factory, to check if the factory is within
specifications. Those 41 widgets have a variance of 3.18mm^{2} in
length.
 State the null and alternative hypotheses, both in words and in
notation. [3]
H_{0}: variance σ^{2} ≤ 2.5
H_{A}: variance σ^{2} > 2.5
 What statistical test is be appropriate to test the hypothesis?
Should it be 1tailed or 2tailed? [2]
χ^{2} (chisquared), 1tailed.
 Run the test (either pvalue or classical approach) and
draw a conclusion. [4]
χ^{2} = 50.88, df = 40.
Pvalue: p > 0.10 (actual p = 0.116)
Classical: χ^{2}* = 55.8.
 Interpret your conclusion in the context of the original
research question.
Please use complete English sentences. [2]
We fail to reject the null hypothesis: there is insufficient evidence
to show that the factory is out of spec.
 What assumptions did you rely upon in conducting the test?
[2]
Random sampling; length of widgets is normally distributed.

Drug company "Faizer" claims their antidepressant is preferred by 30 out of
40 physicians. In response, drug company "Vonartis" claims their
antidepressant is preferred by 60 out of 70 physicians. Is Vonartis' drug
significantly more preferred than Faizer's?
 State the null and alternative hypotheses, both in words and in
notation. [3]
H_{0}: binomial proportion p_{V} ≤ p_{F}
H_{A}: binomial proportion p_{V} > p_{F}
 What statistical test is be appropriate to test the hypothesis?
Should it be 1tailed or 2tailed? [2]
Comparing two independent proportions, 1tailed.
 Run the test (either pvalue or classical approach) and
draw a conclusion. [4]
SE_{V} = sqrt(pq/n) = sqrt(0.857*0.143/70) = 0.04182
SE_{F} = sqrt(pq/n) = sqrt(0.75*0.25/40) = 0.068465
SE_{p1p2} = sqrt(0.857*0.143/70 + 0.75*0.25/40) = 0.08023
z = ( (p'_{V}  p'_{F})  0 ) / SE
= (0.857  0.75) / 0.08023 = 1.335
Pvalue: 0.05 < p < 0.10 (actual p = 0.09)
Classical: z* = 1.65
 Interpret your conclusion in the context of the two drug companies.
Please use complete English sentences. [2]
Fail to reject H_{0}, insufficient evidence to show
that more physicians prefer Vonartis' drug than Faizer's.
 What assumptions did you rely upon in conducting the test?
Are the assumptions met? Why? [2]
Random sampling of physicians (this is a big assumption here!),
np = 30 > 5, nq = 10 > 5, np = 60 > 5, nq = 10 > 5.

"Faizer" and "Vonartis" also produce competing blood glucose monitors.
An independent lab obtains one glucose monitoring
device from each company. A single sample of blood is tested 15 times in each
company's glucose monitoring device. Faizer's device yields a standard
deviation of 0.35 mmol/L; Vonartis' device yields a standard deviation of
0.28 mmol/L. Is there a difference in the precision of the two companies'
devices?
 State the null and alternative hypotheses, both in words and in
notation. [3]
H_{0}: σ_{F} = σ_{V}, or
σ_{F}^{2} / σ_{V}^{2} = 1
H_{A}: σ_{F} ≠ σ_{V}, or
σ_{F}^{2} / σ_{V}^{2} ≠ 1
 What statistical test is be appropriate to test the hypothesis?
Should it be 1tailed or 2tailed? [2]
Ftest comparing two variances, 2tailed.
 Run the test (either pvalue or classical approach) and
draw a conclusion. [5]
F = 0.35^{2} / 0.28^{2} = 1.5625, df = (14, 14)
2tailed pvalue: p > 0.10 (actual p = 0.414)
Classical: F*(12,14,0.025) = 3.05, F*(15,14,0.025) = 2.95
 Interpret your conclusion in the context of the two drug companies.
Please use complete English sentences. [2]
Fail to reject H0: no significant difference in precision between
the two companies' blood glucose monitors.
 What assumptions did you rely upon in conducting the test? [2]
Random sampling of blood, normal distribution of blood glucose levels.

Human betaendorphin (HBE) is a hormone secreted by the pituitary gland under
conditions of stress (like exams!). Suppose we wish to determine whether
blood concentration of HBE (pg/mL) is different for men who
exercise regularly as compared with men who do not exercise
regularly.
 State the null and alternative hypotheses, both in words and in
notation. [3]
[H_{0}: μ_{ex} = μ_{no}:
HBE levels are the same for both groups.
H_{A}: μ_{ex} ≠ μ_{no}:
HBE levels are different for the two groups.
 What statistical test is be appropriate to test the hypothesis?
Should it be 1tailed or 2tailed? [2]
ttest on independent groups, 2tailed.
 Data for this experiment are given below. Sketch boxplots
for the data, on a common axis (number line). [4]
 Mean:  SD: 
Exercisers: 
60  58  62  49 
51  58  54 
56  4.7958 
Nonexercisers: 
41  37  51  60 
28  35  
42  11.6276 
 Run the test (either pvalue or classical approach) and
draw a conclusion. [4]
SE_{E} = 1.8127, SE_{N} = 4.7469, SE = 5.0812.
mean diff = 14, so t = 2.755.
df = min(n_{1}, n_{2})  1 = 5 (real df = 6.45).
Pvalue: Twotailed: 0.02 < p < 0.05 (real p = 0.0401)
Classical: t* = 2.57
 Interpret your conclusion in the context of the original
research question.
Please use complete English sentences. [2]
Reject H_{0},
HBE levels are different for exercisers as compared to nonexercisers.
 What assumptions did you rely upon in conducting the test?
[2]
Random sampling on both groups; HBE levels are normally
distributed in both groups; variance of HBE levels is similar in both
groups.