Deck 3: Reasoning From Sample to Population

Question 1

An unbiased estimator achieves what sort of property?

A) Its variance is the lowest of all possible estimators.
B) Its mean is equal to the population parameter it is used to estimate.
C) Its loss function is the squared loss function.
D) It will always have an equal number of positive and negative errors.

Answer 1

B

Question 2

Suppose a sample of five econometrics students' heights were: 69 inches, 73 inches, 65 inches, 67 inches, and 71 inches. The standard deviation of the heights will be in what units?

A) Standard units
B) Z-score units
C) Inches
D) Squared inches

Answer 2

C

Question 3

When calculating the sample variance of a random sample, you divide the sum of the squared deviations (from the sample mean) by N - 1 instead of N to ensure the estimator achieves what property?

A) Efficiency
B) Unbiasedness
C) Minimum variance
D) Squared loss

Answer 3

B

Question 4

If the distribution of one random variable does not depend on the realization of another, the two random variables are considered what?

A) Correlated
B) Identically distributed
C) i.i.d.
D) Independent

Question 5

In a broad sense, the role of a confidence interval for the population mean is meant to accurately portray what?

A) If you take a random sample, the sample mean is a biased estimate of the population mean.
B) Different population distributions have different population means.
C) The uncertainty involved with observing a sample and not the entire population.
D) None of the answers is correct.

Question 6

A range of values such that there is a specified probability that they contain a population parameter is known as a(n):

A) unbiased estimator.
B) sample mean.
C) confidence interval.
D) random sample.

Question 7

The sample standard deviation is related to the sample variance through what functional form?

A) Logarithm
B) Linear
C) Exponential
D) Square root

Question 8

As long as your sample is large enough, you don't have to worry about using the sample standard deviation in place of the unknown population standard deviation in constructing a confidence interval because ________.

A) for low degrees of freedom, the t-distribution is similar to the standard normal distribution
B) for all degrees of freedom, the t-distribution is bell-shaped
C) for a large sample, the t-distribution is similar to the standard normal distribution
D) the sample standard deviation is an efficient estimator of the population standard deviation

Question 9

Suppose a sample of five retail stores' monthly profits are: $4,000, $7,000, $5,000, $3,000, and $1,000. What will the sample variance of stores profits be?

A) $20 million
B) $4 million
C) $10 million
D) $5 million

Question 10

For a random variable that can take values from zero to 10, what would be the maximum sample variance that could be observed from a sample of two observations?

A) 100
B) 0
C) 50
D) Not enough information to be determined.

Question 11

Suppose you had a random sample of 50 observations with a sample mean of 10 and sample standard deviation of 5. Suppose another observation is observed that has a value of 10. How will the sample mean change from the original sample?

A) It will not change.
B) It will increase.
C) It will decrease.
D) Cannot be determined by the information given.

Question 12

As the size of the random sample gets larger, what happens to the standard deviation of the distribution for the sample mean?

A) It gets larger.
B) It gets smaller.
C) It becomes normal.
D) It goes to 1.

Question 13

What sample statistic is not likely to be used to construct a confidence interval of a population mean?

A) Sample mean
B) Sample standard deviation
C) Number of observations
D) Sample range

Question 14

As the size of a random sample gets larger, what does the distribution of the sample mean begin to resemble?

A) The underlying population distribution
B) A t-statistic
C) A normal distribution
D) None of the answers is correct.

Question 15

Suppose you send out 350 surveys to random sample of all past customers (your target population) asking them to report their level of satisfaction with your product. Of the 350, you used the 112 that responded to the survey to construct a confidence interval for the population "satisfaction score." What might be a potential problem with this confidence interval?

A) Predictable-world bias
B) Satisfaction scores are inherently variable.
C) There is likely to be skewness in the sample.
D) The sample you're using is not a random sample from the target population.

Question 16

A confidence interval can be constructed for which of the following population parameters?

A) Population mean
B) Population variance
C) Population standard deviation
D) All of the answers are correct.

Question 17

Which sample statistic is going to dictate the center of a confidence interval for the population mean?

A) Sample mean
B) Sample standard deviation
C) Number of observations
D) Sample range

Question 18

A random variable that takes on an (uncountable) infinite number of values is known as what type of random variable?

A) Discrete
B) Continuous
C) Population parameter
D) Probability density function

Question 19

A random variable that can only take on a countable number of values is known as what type of random variable?

A) Discrete
B) Continuous
C) Population parameter
D) Probability density function

Question 20

Suppose a sample of five econometrics students' heights were: 69 inches, 73 inches, 65 inches, 67 inches, and 71 inches. What is the sample mean of the height of the students?

A) 71 inches
B) 68 inches
C) 68.5 inches
D) 69 inches

Question 21

Which of the following are you assuming is true to calculate the p-value of a test statistic?

A) A selected sample
B) An efficient estimator
C) The null hypothesis
D) The test statistic is normally distributed in small samples.

Question 22

In evaluating the credibility of a null hypothesis, it is standard to construct a single value derived from the sample which is known as the:

A) sample standard deviation.
B) alternative hypothesis.
C) test statistic.
D) random sample.

Question 23

A hypothesis to be tested using a data sample is known as:

A) the null hypothesis.
B) the true hypothesis.
C) the unbiased hypothesis.
D) the population hypothesis.

Question 24

For the same sample, the 95% confidence interval will have what relation to the 99% confidence interval?

A) It will be larger.
B) It will be smaller.
C) They will not overlap.
D) They will be exact complements of each other.

Question 25

To conduct a hypothesis test at the 95% percent confidence interval the maximum p-value that you would be willing to (accept) fail to reject the null hypothesis would be?

A) 0.95
B) 0.95/2 = 0.475
C) 0.05
D) 0.01

Question 26

Suppose you have a random sample of employees in your company and their tenure. The sample mean of this sample is 4.2 years and the sample standard deviation is 4.5 years. How would knowing that the random sample was of size 100 instead of 60 change the 90% confidence interval for the population mean of employee tenure?

A) The 90% confidence interval will be larger for 100 than 60.
B) The 90% confidence interval will be smaller for 100 than 60.
C) The 90% confidence interval will be biased for 60.
D) The 90% confidence interval will be biased for 100.

Question 27

If you were building a hypothesis test to determine whether or not the price elasticity of demand for your product is -3.0, which of the following would be a natural null hypothesis?

A) H₀: Price Elasticity of Demand = 0
B) H₀: Price Elasticity of Demand = 1
C) H₀: Price Elasticity of Demand = -3.0
D) H₀: Price Elasticity of Demand = -1

Question 28

Suppose you have a random sample of 2179 credit scores from a population of mortgage applicants with a sample mean of 620 and known population standard deviation of 70, and would like to calculate the 99% confidence interval of the population mean credit score. Which of the following would be the correct construction?

A) 620 ± 1.65 $\left( \frac { 70 } { \sqrt { 2179 } } \right)$
B) 2179 ± 1.96 $\left( \frac { 70 } { \sqrt { 620 } } \right)$
C) 620 ± 2.58 $\left( \frac { 70 } { \sqrt { 2179 } } \right)$
D) 620 ± 2.58 $\frac { 70 } { 2179 }$

Question 29

If the t-stat for a hypothesis test (with a two-sided alternative) comes back as 2.6, what would be the appropriate conclusion to draw regarding the null hypothesis under a 95% confidence level?

A) Fail to reject the null hypothesis.
B) Reject the null hypothesis.
C) Reject the null hypothesis 5 percent of the time.
D) Not enough information.

Question 30

In principle, why are t-statistics and critical values not as useful in the practical construction of confidence intervals in most applications?

A) Their use is only valid in large samples.
B) Their use is only valid in small samples.
C) They are not often found in most software applications.
D) In large samples (i.e., large degrees of freedom) the t-distribution and standard normal distribution are very similar.

Question 31

Which of the following is not a required assumption in the reasoning behind constructing a hypothesis test of a population mean?

A) The population mean is K.
B) The size of the sample is sufficiently large .
C) The population distribution is normal.
D) The data sample is random.

Question 32

Suppose you have a random sample of 2,179 credit scores from a population of mortgage applicants with a sample mean of 620 and sample standard deviation of 70, and would like to determine if this is sufficient enough to rule out that the population mean is not 610. Which of the following objects would you calculate to make this decision?

A) T-stat
B) Sample range
C) Normal distribution
D) Consistent estimator

Question 33

Suppose you have a random sample of 200 students' GMAT scores that have a sample mean of 700 and sample standard deviation of 50, and would like to calculate the 90 percent confidence interval of the population mean. Which of the following would be the correct construction?

A) 700 ± 1.65 $\left( \frac { 50 } { \sqrt { 200 } } \right)$
B) 700 ± 1.96 $\left( \frac { 50 } { \sqrt { 200 } } \right)$
C) 700 ± 1.65 $\left( \frac { 50 } { 200 } \right)$
D) 700 ± 1.96 $\left( \frac { 50 } { 200 } \right)$

Question 34

The t-stat, a specific instance of a test statistic, is used to evaluate a null hypothesis concerning which population parameter of a random variable?

A) The variance
B) The r-squared
C) The size of the sample
D) The mean

Question 35

Suppose you have a random sample of 2,179 credit scores from a population of mortgage applicants with a sample mean of 620 and sample standard deviation of 70, and would like to calculate the t-stat for the null hypothesis that the population mean is 610. Which of the following is the correct construction of the t-stat?

A) $\frac { 620 - 610 } { 70 / \sqrt { 2179 } }$
B) $\frac { 620 - 610 } { 70 }$
C) $\frac { 620 } { 70 / \sqrt { 2179 } }$
D) $\frac { 610 - 620 } { 70 / \sqrt { 2179 } }$

Question 36

Suppose you have a random sample of 21 credit scores from a population of mortgage applicants with a sample mean of 620 and sample standard deviation of 70, and would like to calculate the 99 percent confidence interval of the population mean credit score. Which of the following would be the correct construction?

A) 620 ± 2.58 $\left( \frac { 70 } { \sqrt { 21 } } \right)$
B) 620 ± 4 $\left( \frac { 70 } { \sqrt { 21 } } \right)$ , because 4 is the t-distribution critical value.
C) 620 ± 4 (70), because 4 is the t-distribution critical value.
D) None of these choices are correct.

Question 37

A convenient way to modify your degree of support for a decision on a hypothesis test is to calculate the ________ in conjunction with test statistic?

A) P-value
B) Standard error
C) Sample standard deviation
D) Unbiased estimator

Question 38

Over many trials, the 99% confidence interval will not include the true population parameter what percent of the time?

A) 1 percent
B) 99 percent
C) 0 percent
D) 0.5 percent

Question 39

The probability of attaining a test statistic at least as extreme as the one that was observed is known as the:

A) extreme value statistic.
B) first order statistic.
C) standardized z score.
D) p-value.

Question 40

Assuming you are testing a null hypothesis (two-tailed) about the population mean and determining whether to reject the null hypothesis based on the t-stat at the 95% confidence level, which t-stat would warrant rejecting the null hypothesis?

A) 1.4
B) -1.4
C) 3
D) 1.6

Question 41

In making an active prediction it is advantageous for the relationship being predicted to be of what sort?

A) Unbiased
B) Efficient
C) Causal
D) Confounding

Question 42

The step that requires the use of inductive reasoning when making an active prediction from a sample of data is:

A) the creation of the test statistic.
B) the creation of the p-value.
C) determining the population parameter from a sample.
D) the null hypothesis.

Question 43

In making active predictions based on sample data it requires the use of what sort of reasoning?

A) Deductive reasoning only
B) Inductive reasoning only
C) Always deductive reasoning and only inductive reasoning when determining if an estimator is unbiased
D) None of the answers is correct.

Question 44

Can a t-stat be negative for the hypothesis test of the population mean of the heights of econometrics students (which will always be positive)?

A) No.
B) Only if the null hypothesized value is positive.
C) Only if the null hypothesized value if negative.
D) None of the answers is correct.

Question 45

To use the normal distribution to calculate the p-value of a hypothesis test involving the population mean, which assumption is not necessary?

A) The sample is random.
B) The null hypothesis is zero.
C) The sample is "large" enough .
D) None of the answers is correct.

Question 46

In making an active prediction that using a large banner advertisement will increase click-through rates based on a sample of data, what is not an appropriate criticism that someone might have for your prediction?

A) That your sample was not randomly selected.
B) That you do not have a large enough sample.
C) That the underlying population distribution is not normal.
D) That you used a sample of data to make a prediction about the population.

Question 47

If one is attempting to make a prediction on how much sales will increase in the event of a price discount of 10%, which step will not use deductive reasoning in conducting the prediction?

A) Calculating the p-value
B) Calculating the test statistic
C) Determining the test statistic's distribution
D) None of the answers is correct.

Question 48

For a given set of sample statistics, changing the null hypothesized value (K) for a population mean changes everything except for what?

A) P-value
B) T-stat
C) Rejection region
D) Sample standard deviation

Question 49

Suppose you have a random sample of employees in your company and their tenure. The sample mean of this sample is 4.2 years and the sample standard deviation is 4.5 years. How would knowing that the random sample was of size 100 instead of 60 change the p-value of the hypothesis test?

A) The p-value will be larger for 100 than 60.
B) The p-value will be smaller for 100 than 60.
C) The p-value will be biased for 60.
D) The p-value will be biased for 100.

Question 50

If you conduct a hypothesis test under the scenario in which you know the population standard deviation, which of the following will not affect the size of the p-value of the hypothesis test?

A) Sample mean
B) Null hypothesis population mean
C) Sample standard deviation
D) Whether the hypothesis is one or two sided