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 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.

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

The accompanying table lists the height (STUDHGHT)in inches and weight (WEIGHT) in pounds of five college students.Calculate the correlation coefficient. STUDHGHT WEIGHT 74 165 73 165 72 145 68 155 66 140

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. $r _ { X Y } = \frac { s _ { X Y } ^ { 2 } } { s _ { X } ^ { 2 } s _ { Y } ^ { 2 } }$ .

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 "=".

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. $1.96$ .

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.

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 $\bar { Y }$ .

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?

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 $1.96 \times S E ( \hat { p } )$ .

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.

(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 $\left. E ( \hat { \mu } ) \text { in } E ( \hat { \mu } - \mu ) ^ { 2 } . \right)$