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.

Accepted Answer

An unbiased estimator is one whose expected value is equal to the population parameter being estimated. In other words, on average, it doesn't overestimate or underestimate the parameter. This is an important property to have in an estimator as it ensures that the estimates are not systematically biased. While options A, C, and D might also describe desirable properties for certain estimators, they are not specific to unbiased estimators.

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

Accepted Answer

The standard deviation of the heights will be in the same units as the original data (inches). The other options (standard units, z-score units, and squared inches) do not make sense in this context.

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

Accepted Answer

Dividing by N - 1 instead of N when calculating the sample variance ensures that the estimator is unbiased. This adjustment compensates for the fact that the sample mean is used in the calculation, which tends to underestimate the population variance when N is used in the denominator.

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

Accepted Answer

If the distribution of one random variable does not depend on the realization of another, the two random variables are considered independent. Correlation refers to the strength and direction of a linear relationship between two variables, whereas identically distributed (i.i.d.) refers to a sequence of random variables that are all distributed in the same way. While the two variables may be identically distributed, they may not necessarily be 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.

Accepted Answer

A confidence interval for the population mean is meant to accurately portray the uncertainty involved with observing a sample and not the entire population. This means that the interval provides a range of values within which the true population mean is likely to lie, given the observed sample mean and standard deviation. Option A is true but not the main purpose of a confidence interval. Option B is also true but not the main purpose of a confidence interval. Option D is incorrect as option C 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.

Accepted Answer

A confidence interval provides a range of values that is likely to contain the true population parameter with a certain probability or level of confidence.

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

Accepted Answer

The sample standard deviation (s) is the square root of the sample variance (s^2). This relationship is given by the formula s = sqrt(s^2). Therefore, the functional form that relates the sample standard deviation to the sample variance is the 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

Accepted Answer

As per the Central Limit Theorem, for a large enough sample (usually considered when the sample size is greater than or equal to 30), the distribution of the sample means becomes approximately normal regardless of the shape of the population distribution. Therefore, when the sample size is large enough, the t-distribution is very similar to the standard normal distribution, and hence, we can use the sample standard deviation in place of the unknown population standard deviation while constructing the confidence interval.

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

Accepted Answer

To find the sample variance, we first need to find the sample mean:
Mean = ($4,000 + $7,000 + $5,000 + $3,000 + $1,000) / 5 = $4,000

Next, we calculate the deviation of each data point from the mean:
($4,000 - $4,000) = 0
($7,000 - $4,000) = $3,000
($5,000 - $4,000) = $1,000
($3,000 - $4,000) = -$1,000
($1,000 - $4,000) = -$3,000

We square each deviation:
0^2 = 0
3,000^2 = 9,000,000
1,000^2 = 1,000,000
(-1,000)^2 = 1,000,000
(-3,000)^2 = 9,000,000

We sum the squared deviations:
0 + 9,000,000 + 1,000,000 + 1,000,000 + 9,000,000 = 20,000,000

Finally, we divide by n-1 (where n is the sample size):
20,000,000 / 4 = 5,000,000

Therefore, the sample variance is $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.

Accepted Answer

The formula for finding the maximum sample variance is ((max-min)^2)/4. In this case, the maximum value is 10 and the minimum value is 0, so the maximum sample variance would be ((10-0)^2)/4 = 50. Choice C is the only option that matches this value.

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.

Accepted Answer

The new observation has a value of 10, which is the same as the original sample mean. Therefore, when this new observation is added to the original sample, the sample mean will not change.

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.

Accepted Answer

As the sample size gets larger, the standard deviation of the distribution for the sample mean gets smaller. This is due to the Central Limit Theorem, which states that as the sample size increases, the distribution of the sample mean approaches a normal distribution with a mean equal to the population mean and a standard deviation equal to the population standard deviation divided by the square root of the sample size. Therefore, as the sample size increases, the variability of the sample mean decreases, resulting in a smaller standard deviation.

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

Accepted Answer

Sample range is not likely to be used to construct a confidence interval of a population mean as it only provides information about the spread of the data, not the central tendency. The sample mean, sample standard deviation, and number of observations are all commonly used to calculate a confidence interval for a population mean.

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.

Accepted Answer

According to the Central Limit Theorem, as the sample size increases, the distribution of the sample mean becomes approximately normal regardless of the underlying population distribution.

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.

Accepted Answer

The potential problem with this confidence interval is that the sample used is not a random sample from the target population. This limits the generalizability of the results to the entire population and may introduce bias into the sample.

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.

Accepted Answer

Confidence intervals can be constructed for various population parameters, including the population mean, population variance, and population standard deviation, using different statistical methods and distributions (e.g., t-distribution for the mean, chi-square distribution for the variance).

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

Accepted Answer

The sample mean is the best choice for dictating the center of a confidence interval for the population mean as it represents the average value of the sample, which is a good estimate for the population mean.

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

Accepted Answer

A continuous random variable takes on an infinite number of values within a range or interval, and these values are uncountable. Discrete random variables, on the other hand, take on a finite or countable number of values. Population parameters are fixed numerical values that describe characteristics of a population, while probability density functions describe the probability distribution of a continuous random variable.

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

Accepted Answer

A discrete random variable can only take on a countable number of values, while a continuous random variable can take on an uncountable number of values. Population parameter refers to a characteristic of a population, and probability density function is a function used to describe the likelihood of different outcomes in a continuous random variable.

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

Accepted Answer

The answer of Suppose a sample of five econometrics students'...

Quiz 3: Reasoning From Sample to Population

An unbiased estimator achieves what sort of property?

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?

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?

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

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

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

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

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 ________.

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?

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?

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?

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

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

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

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

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

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

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

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?