Question 1

For each of the accompanying scatterplots for several pairs of variables, indicate
whether you expect a positive or negative correlation coefficient between the two
variables, and the likely magnitude of it (you can use a small range).
(a)

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The answer of For each of the accompanying scatterplots for...

Question 2

An estimate is

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A) efficient if it has the smallest variance possible. 
B) a nonrandom number. 
C) unbiased if its expected value equals the population value. 
D) another word for estimator.

Question 3

The t-statistic has the following distribution: a. standard normal distribution for $n < 15$
b. Student $t$ distribution with $n$-1 degrees of freedom regardless of the distribution of the $Y$.
c. Student $t$ distribution with $n$-1 degrees of freedom if the $Y$ is normally distributed.
d. a standard normal distribution if the sample standard deviation goes to zero.

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The answer of The t-statistic has the following distribution: a....

Question 4

The following types of statistical inference are used throughout econometrics, with the exception of

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A) confidence intervals. 
B) hypothesis testing. 
C) calibration. 
D) estimation.

Question 5

With i.i.d. sampling each of the following is true except

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A)  $E ( \bar { Y } ) = \mu _ { \gamma }$ 
B)  $\operatorname { var } ( \bar { Y } ) = \sigma _ { Y } ^ { 2 } / n$ 
C)  $E ( \bar { Y } ) < E ( Y )$ 
D) $\bar { Y }$  is a random variable.

Question 6

An estimator is

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A) an estimate. 
B) a formula that gives an efficient guess of the true population value. 
C) a random variable. 
D) a nonrandom number.

Question 7

The accompanying table lists the height (STUDHGHT)in inches and weight (WEIGHT)
in pounds of five college students.Calculate the correlation coefficient. $$\begin{array}{l}
\begin{array} { l l } 
\text { STUDHGHT}&\text { WEIGHT }\
\hline \hline 74 & 165 \
73 & 165 \
72 & 145 \
68 & 155 \
66 & 140
\end{array}
\end{array}$$

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The answer of The accompanying table lists the height (STUDHGHT)in...

Question 8

The following statement about the sample correlation coefficient is true. a. $- 1 \leq r _ { X Y } \leq 1$.
b. $r _ { X Y } ^ { 2 } \stackrel { p } { \rightarrow } \operatorname { corr } \left( X _ { i } , Y _ { i } \right)$.
c. $\left| r _ { X Y } \right| < 1$.
d. $r _ { X Y } = \frac { s _ { X Y } ^ { 2 } } { s _ { X } ^ { 2 } s _ { Y } ^ { 2 } }$.

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The answer of The following statement about the sample correlation...

Question 9

A type II error a. is typically smaller than the type I error.
b. is the error you make when choosing type II or type I.
c. is the error you make when not rejecting the null hypothesis when it is false.
d. cannot be calculated when the alternative hypothesis contains an "=".

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The answer of A type II error a. is typically...

Question 10

The t-statistic is defined as follows: a. $t = \frac { \bar { Y } - \mu _ { Y , 0 } } { \frac { \sigma _ { Y } ^ { 2 } } { n } }$.
b. $t = \frac { \bar { Y } - \mu _ { Y , 0 } } { S E ( \bar { Y } ) }$.
c. $t = \frac { \left( \bar { Y } - \mu _ { Y , 0 } \right) ^ { 2 } } { S E ( \bar { Y } ) }$.
d. $1.96$.

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The answer of The t-statistic is defined as follows: a....

Question 11

The sample covariance can be calculated in any of the following ways, with the exception of: a. $\frac { 1 } { n - 1 } \sum _ { i = 1 } ^ { n } \left( X _ { i } - \bar { X } \right) \left( Y _ { i } - \bar { Y } \right)$.
b. $\frac { 1 } { n - 1 } \sum _ { i = 1 } ^ { n } X _ { i } Y _ { i } - \frac { n } { n - 1 } \bar { X } \bar { Y }$.
c. $\frac { 1 } { n } \sum _ { i = 1 } ^ { n } \left( X _ { i } - \mu _ { X } \right) \left( Y _ { i } - \mu _ { Y } \right)$.
d. $r _ { X Y } s _ { Y } s _ { Y }$, where $r _ { X Y }$ is the correlation coefficient.

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The answer of The sample covariance can be calculated in...

Question 12

Consider the following alternative estimator for the population mean: $$\tilde { Y } = \frac { 1 } { n } \left( \frac { 1 } { 4 } Y _ { 1 } + \frac { 7 } { 4 } Y _ { 2 } + \frac { 1 } { 4 } Y _ { 3 } + \frac { 7 } { 4 } Y _ { 4 } + \ldots + \frac { 1 } { 4 } Y _ { n - 1 } + \frac { 7 } { 4 } Y _ { n } \right)$$
Prove that $\tilde { Y }$ is unbiased and consistent, but not efficient when compared to $\bar { Y }$.

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The answer of Consider the following alternative estimator for the...

Question 13

You have collected weekly earnings and age data from a sub-sample of 1,744
individuals using the Current Population Survey in a given year.
.
(a)Given the overall mean of $434.49 and a standard deviation of $294.67, construct a
99% confidence interval for average earnings in the entire population.State the
meaning of this interval in words, rather than just in numbers.If you constructed a 90%
confidence interval instead, would it be smaller or larger? What is the intuition?

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The answer of You have collected weekly earnings and age...

Question 14

L Let $p$ be the success probability of a Bernoulli random variable $Y$, i.e., $p = \operatorname { Pr } ( Y = 1 )$. It can be shown that $\hat { p }$, the fraction of successes in a sample, is asymptotically distributed $N \left( p , \frac { p ( 1 - p ) } { n } \right)$. Using the estimator of the variance of $\hat { p } , \frac { \hat { p } ( 1 - \hat { p } ) } { n }$, construct a $95 \%$ confidence interval for $p$. Show that the margin for sampling error simplifies to $1 / \sqrt { n }$ if you used 2 instead of $1.96$ assuming, conservatively, that the standard error is at its maximum. Construct a table indicating the sample size needed to generate a margin of sampling error of $1 \% , 2 \% , 5 \%$ and $10 \%$. What do you notice about the increase in sample size needed to halve the margin of error? (The margin of sampling error is $1.96 \times S E ( \hat { p } )$.

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The answer of L Let $p$ be the success probability...

Question 15

Think of at least nine examples, three of each, that display a positive, negative, or no
correlation between two economic variables.In each of the positive and negative
examples, indicate whether or not you expect the correlation to be strong or weak.

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The answer of Think of at least nine examples, three...

Question 16

(Advanced) Unbiasedness and small variance are desirable properties of estimators. However, you can imagine situations where a trade-off exists between the two: one estimator may be have a small bias but a much smaller variance than another, unbiased estimator. The concept of "mean square error" estimator combines the two concepts. Let  $\hat { \mu }$ be an estimator of $\mu$ Then the mean square error (MSE) is defined as follows: $\operatorname { MSE } ( \hat { \mu } ) = E ( \hat { \mu } - \mu ) ^ { 2 } \text {. Prove that } \operatorname { MSE } ( \hat { \mu } ) = \operatorname { bias } ^ { 2 } + \operatorname { var } ( \hat { \mu } ) \text {. }$
(Hint: subtract and add $\left. E ( \hat { \mu } ) \text { in } E ( \hat { \mu } - \mu ) ^ { 2 } . \right)$

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The answer of (Advanced) Unbiasedness and small variance are desirable...

Exam 3: Review of Statistics

For each of the accompanying scatterplots for several pairs of variables, indicate whether you expect a positive or negative correlation coefficient between the two variables, and the likely magnitude of it (you can use a small range). (a)

An estimate is

The following types of statistical inference are used throughout econometrics, with the exception of

With i.i.d. sampling each of the following is true except

An estimator is

A type II error a. is typically smaller than the type I error. b. is the error you make when choosing type II or type I. c. is the error you make when not rejecting the null hypothesis when it is false. d. cannot be calculated when the alternative hypothesis contains an "=".

Think of at least nine examples, three of each, that display a positive, negative, or no correlation between two economic variables.In each of the positive and negative examples, indicate whether or not you expect the correlation to be strong or weak.