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1. Heart rate during laughter. Laughter is often called “the best medicine,” since studies have shown that laughter can reduce muscle tension and increase oxygenation of the blood. In the International Journal of Obesity (January 2007), researchers at Vanderbilt University investigated the physiological changes that accompany laughter. Ninety subjects (18 – 34 years old) watched film clips designed to evoke laughter. During the laughing period, the researchers measured the
heart rate (beats per minute) of each subject with the following summary results: �̅� = 73.5, s = 6. (NOTE: �̅� and s denote the mean and standard deviation, respectively). It is well known that the mean resting heart rate of adults is 71 beats/minute. At ∝ = 0.05, is there sufficient evidence to indicate that the true mean heart rate during laughter exceeds 71 beats/minute?
a. State the null hypothesis (H0): _____________________________
b. State the alternative hypothesis (HA): _____________________________
c. Calculate the test statistics (z): __________________________________
d. Write decision rule: ___________________________________________
e. State the decision (Reject or do not reject H0): _____________________
Last Name: _______________________________________ First Name: _________________________________
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2. Perform least squares estimation using the data from the table below, and write your answers as requested (attach your calculations on separate pages):
X y
0 1
3 7
5 12
Write your answers in the blank provided and show your manual calculations on separate pages (if necessary).
a. Regression equation: �̂� = _________________________________
Complete the following analysis of variance (ANOVA) table (show calculations on a separate page):
Source Degrees of Freedom (d.f.)
Sum of Squares (SS)
Mean Square (MS)
F
Model
Error
Total
b. Root MSE: ________
c. R-Square: ________
i. Interpretation: __________________________________________
__________________________________________
d. R: ________
i. Interpretation: __________________________________________
__________________________________________
e. Estimate the value for y when x = 3.75: ___________________________
Last Name: _______________________________________ First Name: _________________________________
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3. Earnings of Mexican street vendors: Detailed interviews were conducted with over 1,000 street vendors in the city of Puebla, Mexico, in order to study the factors influencing vendors’ incomes (World Development, February 1998). Vendors were defined as individuals working in the street, and included vendors with cars and stands on wheels and excluded beggars, drug dealers, and prostitutes. The research collected data on gender, age, hours worked per day, annual earnings, and education level. A subset of these appear in the table below.
VenNum Earnings Age Hours
21 2841 29 12
53 1876 21 8
60 2934 62 10
184 1552 18 10
263 3065 40 11
281 3670 50 11
354 2005 65 5
401 3215 44 8
515 1930 17 8
633 2010 70 6
677 3111 20 9
710 2882 29 9
800 1683 15 5
914 1817 14 7
997 4066 33 12
a. Write a first-order model for mean annual earnings, E(y), as a function of age (x1) and hours worked (x2).
____________________________________________________________________
Answer the questions below using data from the printout shown on the final page.
b. Find the least squares prediction equation: _________________________________
Last Name: _______________________________________ First Name: _________________________________
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c. Interpret the estimated β coefficients in your model:
i. Β0: ___________________________________________________________
___________________________________________________________
ii. Β1: ___________________________________________________________
d. Conduct a test of the global utility of the model (at ∝ = 0.01). Interpret the result.
___________________________________________________________
___________________________________________________________
e. Find