Deck 3: Review of Statistics

Question 1

A large p -value implies

A) rejection of the null hypothesis.
B) a large t -statistic.
C) a large $\bar { Y } ^ { \text {act } } .$
D) that the observed value $\bar { Y } ^ { a c t }$ is consistent with the null hypothesis.

Answer 1

that the observed value $\bar { Y } ^ { a c t }$ is consistent with the null hypothesis.

Question 2

The standard error of $\bar { Y } , S E ( \bar { Y } ) = \hat { \sigma } _ { \bar { Y } }$
is given by the following formula:

A) $\frac { 1 } { n } \sum _ { i = 1 } ^ { n } \left( Y _ { i } - \bar { Y } \right) ^ { 2 }$
B) $\frac { s _ { Y } ^ { 2 } } { n } .$
C) $S _ { Y }$
D) $\frac { s _ { Y } } { \sqrt { n } }$

Answer 2

$\frac { s _ { Y } } { \sqrt { n } }$

Question 3

An estimator $\hat { \mu } _ { r }$ of the population value $\mu _ { Y }$ is more efficient when compared to another estimator $\tilde { \mu } _ { Y }$ , if

A) $E \left( \hat { \mu } _ { Y } \right) > E \left( \tilde { \mu } _ { Y } \right)$
B) it has a smaller variance.
C) its c.d.f. is flatter than that of the other estimator.
D) both estimators are unbiased, and $\operatorname { var } \left( \hat { \mu } _ { Y } \right) < \operatorname { var } \left( \tilde { \mu } _ { Y } \right)$

Answer 3

both estimators are unbiased, and $\operatorname { var } \left( \hat { \mu } _ { Y } \right) < \operatorname { var } \left( \tilde { \mu } _ { Y } \right)$

Question 4

An estimator $\hat { \mu } _ { Y }$ of the population value $\mu _ { Y }$ is unbiased if

A) $\hat { \mu } _ { y } = \mu _ { y }$
B) $\bar { Y }$ has the smallest variance of all estimators.
C) $\bar { Y } \stackrel { p } { \rightarrow } \mu _ { y }$
D) $E \left( \hat { \mu } _ { Y } \right) = \mu _ { Y }$

Question 5

To derive the least squares estimator $\mu _ { Y }$ , you find the estimator m which minimizes

A) $\sum _ { i = 1 } ^ { n } \left( Y _ { i } - m \right) ^ { 2 }$
B) $\left| \sum _ { i = 1 } ^ { n } \left( Y _ { i } - m \right) \right|$
C) $\sum _ { i = 1 } ^ { n } m Y _ { i } ^ { 2 }$
D) $\sum _ { i = 1 } ^ { n } \left( Y _ { i } - m \right)$ .

Question 6

An estimator is

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

An estimator $\hat { \mu } _ { Y }$ of the population value $\mu _ { Y }$ is consistent if

A) $\hat { \mu } _ { Y } \stackrel { p } { \rightarrow } \mu _ { Y } .$
B) its mean square error is the smallest possible.
C) Y is normally distributed.
D) $\bar { Y } \stackrel { p } { \rightarrow } 0$

Question 8

When you are testing a hypothesis against a two-sided alternative, then the alternative is written as

Question 9

If the null hypothesis states $H _ { 0 } : E ( Y ) = \mu _ { Y , 0 }$ , then a two-sided alternative hypothesis is

A) $H _ { 1 } : E ( Y ) \neq \mu _ { Y , 0 }$
B) $H _ { 1 } : E ( Y ) \approx \mu _ { Y , 0 }$
C) $H _ { 1 } : \mu _ { y } < \mu _ { y , 0 }$
D) $H _ { 1 } : E ( Y ) > \mu _ { Y , 0 ^ { \circ } }$

Question 10

Among all unbiased estimators that are weighted averages of $Y _ { 1 } , \ldots , Y _ { n } , \bar { Y }$
is

A) the only consistent estimator of $\mu _ { Y }$
B) the most efficient estimator of $\mu _ { Y }$
C) a number which, by definition, cannot have a variance.
D) the most unbiased estimator of $\mu _ { Y }$

Question 11

A type I error is

A)always the same as (1-type II)error.
B)the error you make when rejecting the null hypothesis when it is true.
C)the error you make when rejecting the alternative hypothesis when it is true.
D)always 5%.

Question 12

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

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

Question 13

The p -value is defined as follows:

A) p=0.05 .
B) $\operatorname { Pr } _ { H _ { n } } \left[ \left| \bar { Y } - \mu _ { Y , 0 } \right| > \left| \bar { Y } ^ { a c t } - \mu _ { Y , 0 } \right| \right] .$
C) $\operatorname { Pr } ( z > 1.96 )$
D) $\operatorname { Pr } _ { H _ { 0 } } \left[ \left| \bar { Y } - \mu _ { Y , 0 } \right| < \bar { Y } ^ { \text {act } } - \mu _ { Y , 0 } \mid \right] . .$

Question 14

The power of the test is

Question 15

A scatterplot

Question 16

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

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 17

The critical value of a two-sided t-test computed from a large sample

A)is 1.64 if the significance level of the test is 5%.
B)cannot be calculated unless you know the degrees of freedom.
C)is 1.96 if the significance level of the test is 5%.
D)is the same as the p-value.

Question 18

A type II error

Question 19

The size of the test

A)is the probability of committing a type I error.
B)is the same as the sample size.
C)is always equal to (1-the power of test).
D)can be greater than 1 in extreme examples.

Question 20

An estimate is

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 21

Degrees of freedom

Question 22

The t-statistic has the following distribution:

Question 23

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?

Question 24

U.S.News and World Report ranks colleges and universities annually.You randomly
sample 100 of the national universities and liberal arts colleges from the year 2000
issue.The average cost, which includes tuition, fees, and room and board, is $23,571.49
with a standard deviation of $7,015.52.
(a)Based on this sample, construct a 95% confidence interval of the average cost of
attending a university/college in the United States.

Question 25

The sample covariance can be calculated in any of the following ways, with the exception of:

Question 26

The t-statistic is defined as follows:

Question 27

A manufacturer claims that a certain brand of VCR player has an average life
expectancy of 5 years and 6 months with a standard deviation of 1 year and 6 months.
Assume that the life expectancy is normally distributed.
(a)Selecting one VCR player from this brand at random, calculate the probability of its life
expectancy exceeding 7 years.

Question 28

The standard error for the difference in means if two random variables M and W , when the two population variances are different, is

Question 29

Adult males are taller, on average, than adult females.Visiting two recent American
Youth Soccer Organization (AYSO)under 12 year old (U12)soccer matches on a
Saturday, you do not observe an obvious difference in the height of boys and girls of
that age.You suggest to your little sister that she collect data on height and gender of
children in 4th to 6th grade as part of her science project.The accompanying table shows
her findings. (a)Let your null hypothesis be that there is no difference in the height of females and
males at this age level.Specify the alternative hypothesis.

Question 30

Your textbook defines the correlation coefficient as follows: Another textbook gives an alternative formula: Prove that the two are the same.
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Question 31

The following statement about the sample correlation coefficient is true.

Question 32

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)

Question 33

When the sample size n is large, the 90% confidence interval for $\mu _ { Y }$ is

A) $\bar { Y } \pm 1.96 S E ( \bar { Y } )$
B) $\bar { Y } \pm 1.64 S E ( \bar { Y } )$
C) $\bar{Y} \pm 1.64 \sigma_{Y}$

D) $\bar { Y } \pm 1.96 \text {. }$

Question 34

During the last few days before a presidential election, there is a frenzy of voting
intention surveys.On a given day, quite often there are conflicting results from three
major polls.
(a)

Question 35

Your textbook defined the covariance between X and Y as follows:

Question 36

The power of the test

Question 37

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.

Question 38

(a)Sketch the two distributions under the null hypothesis and the alternative hypothesis.

Question 39

The correlation coefficient

A)lies between zero and one.
B)is a measure of linear association.
C)is close to one if X causes Y.
D)takes on a high value if you have a strong nonlinear relationship.

Question 40

The formula for the sample variance is

A) $s _ { Y } ^ { 2 } = \frac { 1 } { n - 1 } \sum _ { i = 1 } ^ { n } \left( Y _ { i } - \bar { Y } \right)$
B) $s _ { Y } ^ { 2 } = \frac { 1 } { n - 1 } \sum _ { i = 1 } ^ { n } \left( Y _ { i } - \bar { Y } \right) ^ { 2 }$
C) $s _ { Y } ^ { 2 } = \frac { 1 } { n - 1 } \sum _ { i = 1 } ^ { n } \left( Y _ { i } - \mu _ { Y } \right) ^ { 2 }$
D) $s _ { Y } ^ { 2 } = \frac { 1 } { n - 1 } \sum _ { i = 1 } ^ { n - 1 } \left( Y _ { i } - \bar { Y } \right) ^ { 2 }$

Question 41

Imagine that you had sampled 1,000,000 females and 1,000,000 males to test whether
or not females have a higher IQ than males.IQs are normally distributed with a mean of
100 and a standard deviation of 16.You are excited to find that females have an
average IQ of 101 in your sample, while males have an IQ of 99.Does this difference
seem important? Do you really need to carry out a t-test for differences in means to
determine whether or not this difference is statistically significant? What does this
result tell you about testing hypotheses when sample sizes are very large?

Question 42

Your textbook suggests using the first observation from a sample of n as an estimator of
the population mean.It is shown that this estimator is unbiased but has a variance of
σ2 , which makes it less efficient than the sample mean.Explain why this estimator is
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not consistent.You develop another estimator, which is the simple average of the first
and last observation in your sample.Show that this estimator is also unbiased and show
that it is more efficient than the estimator which only uses the first observation.Is this
estimator consistent?

Question 43

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Question 44

Some policy advisors have argued that education should be subsidized in developing
countries to reduce fertility rates.To investigate whether or not education and fertility
are correlated, you collect data on population growth rates (Y)and education (X)for 86
countries.Given the sums below, compute the sample correlation:

Question 45

Your textbook states that when you test for differences in means and you assume that
the two population variances are equal, then an estimator of the population variance is
the following "pooled" estimator: 2
Explain why this pooled estimator can be looked at as the weighted average of the two
variances.

Question 46

The accompanying table lists the height (STUDHGHT)in inches and weight (WEIGHT)
in pounds of five college students.Calculate the correlation coefficient.

Question 47

(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 be an estimator of Then the mean square error (MSE) is defined as follows:
(Hint: subtract and add

Question 48

Consider two estimators: one which is biased and has a smaller variance, the other
which is unbiased and has a larger variance.Sketch the sampling distributions and the
location of the population parameter for this situation.Discuss conditions under which
you may prefer to use the first estimator over the second one.

Question 49

Your textbook mentions that dividing the sample variance by n -1 instead of n is called
a degrees of freedom correction.The meaning of the term stems from the fact that one
degree of freedom is used up when the mean is estimated.Hence degrees of freedom
can be viewed as the number of independent observations remaining after estimating
the sample mean.
Consider an example where initially you have 20 independent observations on the
height of students.After calculating the average height, your instructor claims that you
can figure out the height of the 20th student if she provides you with the height of the
other 19 students and the sample mean.Hence you have lost one degree of freedom, or
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there are only 19 independent bits of information.Explain how you can find the height
of the 20th student.

Question 50

IQs of individuals are normally distributed with a mean of 100 and a standard deviation
of 16.If you sampled students at your college and assumed, as the null hypothesis, that
they had the same IQ as the population, then in a random sample of size

Question 51

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Question 52

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Question 53

Consider the following alternative estimator for the population mean:

Question 54

When you perform hypothesis tests, you are faced with four possible outcomes
described in the accompanying table.

Question 55

The net weight of a bag of flour is guaranteed to be 5 pounds with a standard deviation
of 0.05 pounds.You are concerned that the actual weight is less.To test for this, you
sample 25 bags.Carefully state the null and alternative hypothesis in this situation.
Determine a critical value such that the size of the test does not exceed 5%.Finding the
average weight of the 25 bags to be 4.7 pounds, can you reject the null hypothesis?
What is the power of the test here? Why is it so low?

Question 56

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