Statistics- STAT200 Homework 11 Due 5 pm, Friday

QUESTION

STAT200 Homework 11 Due 5 pm, Friday, Nov 20,
2015

1.
The data below are concerned with the Leaning
Tower of Pisa. Engineers concerned about the tower’s stability have done
extensive studies of its increasing tilt. The table gives measurements for the
year 1975 to 1987 (coded as 75 – 87). The variable “lean” represents the
difference between where a point on the tower would be if the tower were
straight and where it actually is. The data are coded as tenths of a millimeter
in excess of 2.9 meters, so that 1975 lean, which was 2.9642 meters, appears in
the table as 642. Only the last two digits of the year were entered into the
computer:

Year

75

76

77

78

79

80

81

82

83

Lean

642

644

656

667

673

688

696

698

713

Year

84

85

86

87

Lean

717

725

742

757

(a)
Plot the data. Does the trend in lean over time
appear to be linear?
(b)
What is the equation of the least –squares line?
What percent of the variation in lean is explained by this line?
(c)
Give a 99% confidence interval for the average
rate of change (tenths of a millimeter per year) of the lean.
(d)
The engineers working on the Leaning Tower of
Pisa were most interested in how much the tower would lean if no corrective action
were taken. Use the least-squares equation to predict the tower’s lean in the
year 2000, coded as 100.(Note: The tower was renovated in 2001 to make sure it
does not fall down. )
(e)
What is the 95% prediction interval for the lean
in year 100? What is the 95% confidence interval for the mean response of the
lean then?

2.
In each of the following settings, give a 95%
confidence interval for the coefficient of x2.
(a)
n=26, Ŷ=1.6+6.4 x1+5.7 x2,
SE of b2=3.1
(b)
n=26, Ŷ=1.6+4.8 x1+3.2 x2+5.2
x3, SE of b2=2.2
(c, d) For each of the settings in (a),(b), test
the null hypothesis that the coefficient of x1 is zero versus the two-sides
alternative.
3.
A multiple regression analysis of 78 cases was
performed with 5 explanatory variables. Suppose that SSM=16.5 and SSE=100.8.
(a)
Find the value of the F statistics for testing
the null hypothesis that the coefficients of all the explanatory variables are
zero.
(b)
What are the degrees of freedom of this
statistic?
(c)
Find bounds on the P-value using Table. Show your
work.

4.
Let’s consider developing a model to predict
total score based on the peer review score (PEER), faculty-to-student ratio (FtoS),
and citations-to-faculty ratio (CtoF). Refer
to the attached dataset.
(a)
Generate scatterplots for each pair of the
variables. Do these relationships all look linear?
(b)
Compute the correlation between each pair of the variables.
Are certain explanatory variables more strongly associated with the total
score?

5.
For the same dataset for problem 4, considering a
regression model using all three explanatory variables.
(a)
Write out the statistical model for this
analysis, making sure to specify all assumptions.
(b)
Run the multiple regression model and specify the
fitted regression equation.
(c)
Generate a 95% confidence interval for each
coefficient.
(d)
What percent of the variation does this model
explain? What is the estimate for the standard deviation?
(e)
Is the regression model significantly better than
the intercept only model? Use the ANOVA F-test to test this idea.

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