Download Eco311, Spring 2017 Answer key for problems on variance

Eco311, Spring 2017
Answer key for problems on variance, covariance, and normal distribution.
Complete the problems below before class on Monday 2/6.
Suppose that the distribution of test scores is N(70,100).
1.
What is the probability of a score in the following ranges?
a. Less than 50? F( (50-70)/10)=.02 where F(.) is standard normal cdf.
b. Less than 70? F((70-70)/10)=.5
c. Between 80 and 90? F((90-70)/10)-F((80-70)/10)=.136
2. What is the 95 percent confidence interval (CI) for test scores?
95% CI is obtained by sample mean + 1.96* standard deviations.
95% CI=70 + 1.96*10 = (50.4,89.6)
3. What is the 99 percent confidence interval for test scores?
99% CI is obtained by sample mean ± 2.57*standard deviations
70+ 10*2.57=(44.3, 95.7)
4. Consider the data below for the random variable 𝑋𝑖 which can take on each value with
probability 𝑃𝑖
i
𝑋𝑖
𝑃𝑖
1
2
3
5
10
20
.5
.3
.2
Using the above data, calculate the following:
a.
b.
c.
d.
e.
f.
∑3𝑖=1 𝑋𝑖 = 6
∑3𝑖=1(𝑖 ∗ 𝑋𝑖 ) = 85
𝐸(𝑋𝑖 )=9.5
𝐸(𝑋𝑖2 )=122.5
𝑉𝑎𝑟(𝑋𝑖 )=32.25
𝐸(𝑋𝑖3 ) =1962.5
Eco311, Spring 2017
Answer key for problems on variance, covariance, and normal distribution.
5. The following statistics for f_faminc (family income in 1000s of $) and ss_val (Social Security
Benefits measured in 1000s of $) are based on data from the Current Population Survey in 2010.
Note that the second command gives the variance-covariance between f_faminc and ss_val.
5578 is the variance of f_faminc, 23.81 is the variance of ss_val, and -26.69 is the covariance
between the two variables.
a.
Based on the information provided, what is the correlation between family income and
social security? (Note the bottom table above provides covariances).
b. Using x to denote f_faminc and y to denote ss_val
corr(x,y)=cov(x,y)/((var(x)*var(y))1/2 = -.072
c. Based on the information provided, what is mean and variance of combined income
(f_faminc+ss_val)?
var(x+y)=var(x)+var(y)+2cov(x,y)= 5748.48
d. Assuming that combined income is normally distributed with the mean and variance in
(b), construct a 95% confidence for combined income.
E(x+y)=E(x)+E(y)=74.31
Var(x+y) is given in c
95% CI=74.31 + 1.96 s.d. = 74.31 + 1.96(75.82) = (-74.2.9, 222.91)
6. Assuming that combined income is normally distributed with the mean and variance in (b), what
is the probability that a family would have combined income of more than $250,000 (recall that
the above variables are measured in 1000s of dollars).
1-F((250-74.31)/75.82)=.010 where F(.) is the standard normal cdf.