Question 1 of 8
A researcher on crime performs a regression of crime on unemployment.
Burglaries = a + b unemployment + epsilon
The estimated coefficient b is 0.279 with a standard error of 0.0022. The number of observations
is N=2949.
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Provide the t statistic.
Is the coefficient b significant ? At what confidence level(s)?
Build a 95% confidence interval for b.
Answer given:
t = b/se =0.279/0.0022 = 126.81
The coefficient is significant because |126.81| >1.65, 1.96, and 2.58.
0.279 – 1.96*(0.0022), 0.279 + 1.96*(0.0022)
Question 2 of 8
A researcher on crime performs a regression of crime on unemployment.
Burglaries = a + b unemployment + epsilon
The estimated coefficient b is 0.279 with a standard error of 0.0022. The number of observations
is N=2949. The standard deviation of burglaries is 2313, and the standard deviation of
unemployment is 7985.
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Find the correlation of Burglaries and Unemployment.
Is the correlation of Burglaries and Unemployment significant? At what level(s)?
Answer given:
𝑠
7985
𝑟 = 𝑏 𝑠𝑥 = 0.279 2313=0.963
𝑦
𝑡=
𝑟
√(1 − 𝑟 2 )/(𝑁 − 2)
=
0.963
√(1 − 0.927)/(2949 − 2)
= 193.98
The correlation between burglaries and unemployment is significant at all levels
(90, 95, and 99%) because |t| > 1.65, 1.96, and 2.58.
Question 3 of 8
Consider the example done in the lecture relating the price of a Burger King Entree (the classic
hamburger) to the level of poverty in the ZIP code of the outlet.
The sample size is N=167. The regression is:
Price = a + b * Fraction in Poverty in ZIP code + epsilon
where we find that b = -0.189. The standard error on b is 0.1506. The variable "Fraction in
Poverty in ZIP code" ranges between 0 and 1. The price is in $.
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Find the 95% confidence interval on b without looking at the slides.
Interpret the coefficient b : A 10 percentage point increase in the fraction in Poverty in ZIP
code lowers the price by how many $ ?
How many degrees of freedom should we use for the t score? (do not reply infinity).
Answer given:
CI = (-0.189 – 1.96*0.1506, -0.189 + 1.96*0.1506)
A 10 percentage point increase in the fraction in Poverty in ZIP code lowers the
price by 10*(-0.189)=-1.89$
dF = N – k – 1 = N – 2 = 165
where k is the number of independent variables.
Question 4 of 8
The R squared of a regression is the ratio of the SSE to the TSS. True or False?
Answer given:
False. R squared = ESS/TSS
Question 5 of 8
We reject the null hypothesis that the slope of a regression is 0 at 95% confidence level if and only
if the p value is greater than 0.05. True or False?
Answer given:
False. We reject the null hypothesis that the slope of a regression is 0 at 95%
confidence level if and only if the p value is SMALLER than 0.05.
Question 6 of 8
The t statistic is always of the same sign as the slope coefficient. True or False?
Answer given:
True. T = b/se and we know se is always positive.
Question 7 of 8
The correlation between two variables y and x is statistically different from 0 if and only if the
slope of the regression of x on y is statistically different from 0 (at the same confidence level).
True or False?
Answer given:
True.
𝜎
Note that 𝜌 = 𝜎𝑥 𝛽, hence r = 0 if and only if b = 0.
𝑦
Question 8 of 8
A medical researcher analyzes the relationship between height (cm) and weight (kg), particularly
during the child's and young adult's growth period.
He has a sample of 8462 individuals and the weight variable is bmxwt. The height variable is
bmxht. He regresses weight (y) on height (x).
The output of the regression is as follows:
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Is the slope coefficient significant at 95%? 99%? 90%?
How do you obtain the t statistic in this output without knowing it?
Also find the 95% confidence interval as a function of the other elements of the output.
Same question with the R squared? It is given there, but how would you compute the r squared
from this output?
Answer given:
Yes, the slope coefficient is significant at all levels because the p value is smaller
than 0.01 and the t statistic is greater than 2.58.
t = b/se.
CI = (b – 1.96*se, b + 1.96*se) = (0.9580879 – 1.96*0.0080152, 0.9580879 +
1.96*0.0080152)
R squared = ESS/TSS