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Principles of Data Science

Quantitative Problems

Principles of Data ScienceQuantitative Problems

Quantitative Problems

1.
 
a.
Consider the data in the following table representing world record fastest times for the 100 m sprint and the year in which each occurred, from 1912 through 2002:
Year Time
1912 10.6
1921 10.4
1930 10.3
1936 10.2
1956 10.1
1960 10
1968 9.95
1983 9.93
1988 9.92
1991 9.9
1991 9.86
1994 9.85
1996 9.84
1999 9.79
Table 6.10 100-Meter Spring Records
Using software such as Excel, Python, or similar tools, the regression line can be found. For this data, the linear model would be y ^ = − 0.00769 x + 10.55 , where x is years since 1900. Compute the MAE, MAPE, MSE, and RMSE for this model.
b.
Use the model to predict the world record fastest time for the 100 m spring in 2023. In August of 2023, Noah Lyles ran the 100 m sprint in 9.83 seconds. How does your prediction compare to this value?
2.
If today is a cloudy day, then there is 68% chance of rain tomorrow. If today is not cloudy, then there is only 15% chance of rain tomorrow. Build a discrete logistic regression model based on this data.
3.
Compute the information contained in the events.
a.
Obtaining a head and a tail on a flip of two fair coins
b.
Obtaining all heads on the flip of three fair coins
c.
Rolling 10 fair 6-sided dice and obtaining all 1s
4.
Compute the entropy of the following discrete random variables.
a.
The distribution X whose values are given in the following table.
X 1 2 3 4
p ( x ) 0.7 0.15 0.05 0.05
Table 6.11 Distribution X Values
b.
The binomial distribution with p = 1 / 3 and n = 4 . Recall, the binomial distribution is defined by:
P ( x ) = ( n x ) p x ( 1 − p ) n − x
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