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Regression and Correlation" Exercises": Bio. Stat. & Applied Stat. 2021 2 Semester, DR. Y.M

1. The document contains 5 exercises analyzing regression and correlation from experimental data. It provides the statistical calculations and tests for each exercise, including regression lines, correlation coefficients, and confidence intervals. 2. Exercise 1 analyzes the effect of barbiturate dosage on sleeping time using linear regression to predict sleeping time at a new dosage level. 3. Exercise 2 examines the correlation between psychosis intensity ratings and plasma amphetamine levels, finding no significant correlation. 4. Exercise 3 determines the regression line relating blood pressure to shock intensity in monkeys. It finds a significant relationship between the variables. 5. Exercise 4 analyzes the correlation between medical students' GPAs and exam scores, finding a significant positive

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Omar Saleh
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100 views7 pages

Regression and Correlation" Exercises": Bio. Stat. & Applied Stat. 2021 2 Semester, DR. Y.M

1. The document contains 5 exercises analyzing regression and correlation from experimental data. It provides the statistical calculations and tests for each exercise, including regression lines, correlation coefficients, and confidence intervals. 2. Exercise 1 analyzes the effect of barbiturate dosage on sleeping time using linear regression to predict sleeping time at a new dosage level. 3. Exercise 2 examines the correlation between psychosis intensity ratings and plasma amphetamine levels, finding no significant correlation. 4. Exercise 3 determines the regression line relating blood pressure to shock intensity in monkeys. It finds a significant relationship between the variables. 5. Exercise 4 analyzes the correlation between medical students' GPAs and exam scores, finding a significant positive

Uploaded by

Omar Saleh
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Bio. Stat. & Applied Stat. 2021 2 semester, DR. Y.M.

nd

Regression and Correlation” Exercises”


[1]An experiment was conducted to study the effect of
increasing the dosage of a certain barbiturate on sleeping
time. Three readings were made at each of three dose levels.

Sleeping Time (Hours) 4 6 5 9 8 7 13 11 9


Dosage (μM/kg) 3 3 3 10 10 10 15 15 15
What is the predicted sleeping time for a dose of 12 μM/kg?
Place a 95% confidence interval for β1.

Solution

̂ = 3.38 + 0.495x
𝒚

At x=12

̂ = 3.38 + 0.495(12) = 9.32


𝒚

95% C.I. for β1:

b1 = 0.495 , t0.025 , 7 = 2.365


̂
𝒚 4.865 4.865 4.865 8.33 8.33 8.33 10.805 10.805 10.805
ei = - 1.135 0.135 0.67 -0.33 -1.33 2.195 0.195 -1.805
̂ -
𝒚 0.865
y
SSE=12.495 “using calculator” , MSE = 12.495/7 =1.785

Sxx = 218 “ using calculator “

𝟏.𝟕𝟖𝟓
0.495 ± 2.365 * √
𝟐𝟏𝟖

(0.2809 , 0.709)

1
Bio. Stat. & Applied Stat. 2021 2 semester, DR. Y.M.
nd

[2] In a study of the relationship between amphetamine


metabolism and amphetamine psychosis, six chronic
amphetamine users were given a psychosis intensity rating
score. Plasma amphetamine levels (mg/ml) were also
determined for these patients.
Psychosis Intensity Rating (y) 15 40 45 30 55 30
Plasma Amphetamine (mg/ml)(X) 150 100 200 250 250 500
Determine the correlation coefficient. Is there a
significant correlation between intensity rating and
plasma amphetamine level? Test Ho : 𝜷𝟏 = 𝟎

Solution

r = -0.0472

Test:

Ho : there is no correlation coefficient between intensity


rating and plasma amphetamine

Ha : there is a significant correlation between intensity rating


and plasma amphetamine

𝒏−𝟐 𝟒
Tc = r√ = -0.0472 √ = -0.0945
𝟏−𝒓𝟐 𝟏−(−𝟎.𝟎𝟒𝟕𝟐)𝟐

t0.025,4 = 2.776

Reject Ho if | -0.0945 | ≥ 2.776

Can’t Reject Ho

there is no correlation coefficient between intensity rating


and plasma amphetamine.

2
Bio. Stat. & Applied Stat. 2021 2 semester, DR. Y.M.
nd

Test Ho : 𝜷𝟏 = 𝟎

H0 : 𝜷𝟏 = 𝟎 Ha: 𝜷𝟏 ≠ 𝟎

b0 = 37 , b1 = -0.0047

t0.025,4 = 2.776

MSE= 242.2

Sxx = 97083.33

−𝟎.𝟎𝟎𝟒𝟕
Tc = = -0.094
𝟐𝟒𝟐.𝟐
√
𝟗𝟕𝟎𝟖𝟑.𝟑𝟑

Reject Ho if |-0.094| > 2.776

Can’t Reject Ho , 𝜷𝟏 = 𝟎 , the model is bad fit

[3] An investigator studying the effects of stress on blood


pressure subjected nine monkeys to increasing levels of
electric shock as they attempted to obtain food from a
feeder. At the end of a 2-minute stress period blood
pressure were recorded.
Blood pressure (Y) 125 130 120 150 145 160 175 180 180
Shock Intensity (X) 30 30 30 50 50 50 70 70 70
Determine the regression line relating blood pressure to
intensity of shock. Test Ho : 𝜷𝟏 = 𝟎
Solution

b0 = 85 , b1 = 1.33

̂ = 85 + 1.33x
𝒚

3
Bio. Stat. & Applied Stat. 2021 2 semester, DR. Y.M.
nd

H0 : 𝜷𝟏 = 𝟎 Ha: 𝜷𝟏 ≠ 𝟎

t0.025,7 = 2.365

MSE= 26.2

Sxx = 2400

𝟏.𝟑𝟑
Tc = = 12.73
𝟐𝟔.𝟐
√
𝟐𝟒𝟎𝟎

Reject Ho if | 12.73 | > 2.365

Reject Ho , 𝜷𝟏 ≠ 𝟎 , the model is good fit

[4] The following table gives grade point average at the end
of the first 2 year of basic sciences and scores on National
Boards (NBS) Part 1 for 12 medical students:
GPR (X) 3.35 2.37 3.13 3.10 1.94 3.00 2.85 1.96
2.98 2.55 2.23 1.95
Score (Y) 620 445 445 560 295 570 415 430
560 515 430 435
Determine the correlation between GPR and NBS. Is the
correlation significant? For GBR of 3, what is the predicted
NBS? Test Ho : 𝜷𝟏 = 𝟎 Place 95% confidence interval for β1.
Solution

r=0.7543

Ho : there is no correlation coefficient between GPR and


Score

Ha : there is a significant correlation between GPR and Score

4
Bio. Stat. & Applied Stat. 2021 2 semester, DR. Y.M.
nd

𝒏−𝟐 𝟏𝟎
Tc = r√ = 0.7543 √ = 3.633
𝟏−𝒓𝟐 𝟏−(𝟎.𝟕𝟓𝟒𝟑)𝟐

t0.025,10 = 2.228

Reject Ho if | 3.633 | ≥ 2.228

Reject Ho

There is a significant correlation between GPR and Score

̂ = 131 + 132x
𝒚

At X=3 , 𝒚
̂ = 131 + 132(3) = 527

H0 : 𝜷𝟏 = 𝟎 Ha: 𝜷𝟏 ≠ 𝟎

b0 = 131 , b1 = 132

t0.025,10 = 2.228

MSE= 3858

Sxx = 2.9166

𝟏𝟑𝟐
Tc = = 3.63
𝟑𝟖𝟓𝟖
√
𝟐.𝟗𝟏𝟔𝟔

Reject Ho if | 3.63 | > 2.228

Reject Ho , 𝜷𝟏 ≠ 𝟎 , the model is good fit

C.I.

5
Bio. Stat. & Applied Stat. 2021 2 semester, DR. Y.M.
nd

𝟑𝟖𝟓𝟖
132 ± 2.228 * √
𝟐.𝟗𝟏𝟔𝟔

(50.967 , 213.032)

[5] In a study on the elimination of a certain drug in man, the


following data were recorded:

Time in Hours (X) .5 .5 1 1 2 2 3 3 4 4


Drug Concentration (μg/ml)(Y) .42 .45 .35 .33 .25
.22 .20 .20 .15 .17
What is the predicted drug concentration after 2 hours? Test
Ho : 𝜷𝟏 = 𝟎. Set a 99% confidence interval for β1.
Solution

̂ = 0.43 - 0.0743 x
𝒚

At x=2 , 𝒚
̂ = 0.43 - 0.0743(2) = 0.2814

H0 : 𝜷𝟏 = 𝟎 Ha: 𝜷𝟏 ≠ 𝟎

b0 = 0.43 , b1 = -0.0743

t0.025,8 = 2.306

MSE= 0.001404

Sxx = 16.4

−𝟎.𝟎𝟕𝟒𝟑
Tc = = -8.03
𝟎.𝟎𝟎𝟏𝟒𝟎𝟒
√
𝟏𝟔.𝟒

Reject Ho if | -8.03 | > 2.306


6
Bio. Stat. & Applied Stat. 2021 2 semester, DR. Y.M.
nd

Reject Ho , 𝜷𝟏 ≠ 𝟎 , the model is good fit

t0.005,8 = 1.86

𝟎.𝟎𝟎𝟏𝟒𝟎𝟒
99% C.I. : -0.0743 ± 1.86 * √
𝟏𝟔.𝟒

(-0.0915 , -0.057)

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