Which of the following statements is not true of a discrete probability distribution?

Chapters 1 and 2

Inhaltsverzeichnis Show

Which of the following is not a discrete probability distribution?
Which statement is not true for a probability distribution for a discrete random variable?
What is true about discrete probability distribution?
Which of the following is true of both discrete and continuous probability distributions?

If a researcher uses daily data to examine a particular problem and creates a variable that assigns a numerical value of 1 to Monday observations, what term would best describe this type of number?

	a)	Continuous
	b)	Cardinal
	c)	Ordinal
	d)	Nominal
Correct! This would be a good example of a nominal number, since it does not even produce an ordering - the numbers assigned to each day of the week are entirely arbitrary. There is no sense that Tuesday is "better" than Monday because it is assigned a higher value. We could instead and equally validly have assigned the value 5 to Monday, 4 to Tuesday, and so on. Clearly since the numbers assigned to the days of the week would only comprise 5 values, we would not term it a continuous variable. Incorrect! This would be a good example of a nominal number, since it does not even produce an ordering - the numbers assigned to each day of the week are entirely arbitrary. There is no sense that Tuesday is "better" than Monday because it is assigned a higher value. We could instead and equally validly have assigned the value 5 to Monday, 4 to Tuesday, and so on. Clearly since the numbers assigned to the days of the week would only comprise 5 values, we would not term it a continuous variable. Your answer has been saved.

The price of a house is best described as what type of number?

	a)	Discrete
	b)	Cardinal
	c)	Ordinal
	d)	Nominal
Correct! The price of a house is not a discrete variable because, at least in principle, it can take on any value (limited only by the granularity of the currency it is traded in). It is a cardinal number because the actual numerical values that it takes have a meaning, and for example, a house valued at �500,000 is worth twice as much as one valued at �250,000. Incorrect! The price of a house is not a discrete variable because, at least in principle, it can take on any value (limited only by the granularity of the currency it is traded in). It is a cardinal number because the actual numerical values that it takes have a meaning, and for example, a house valued at �500,000 is worth twice as much as one valued at �250,000. Your answer has been saved.

Which of the following is NOT a feature of continuously compounded returns (i.e. log-returns)?

	a)	They can be interpreted as continuously compounded changes in the prices
	b)	They can be added over time to give returns for longer time periods
	c)	They can be added across a portfolio of assets to give portfolio returns
	d)	They are usually fat-tailed
Correct! Log-returns can indeed be interpreted as continuously compounded changes in the price or index value over time. This is useful since it means we don't have to worry about the compounding frequency. Log-returns can also be added up over time, so that the return over a year is simply the sum of the daily returns for all trading days in that year. Asset returns are usually fat-tailed (leptokurtic), and this is true whether they are measured as log-returns or simple returns. However, log-returns cannot be aggregated across a portfolio to get a portfolio return. This would be possible with simple returns but does not work for log-returns because taking the log is a non-linear transformation process. Therefore the sum of a log is not the same as the sum of a log. In order to calculate portfolio log-returns, it is necessary to calculate the value of the whole portfolio first at each point in time and then to take the log of the portfolio price changes. Incorrect! Log-returns can indeed be interpreted as continuously compounded changes in the price or index value over time. This is useful since it means we don't have to worry about the compounding frequency. Log-returns can also be added up over time, so that the return over a year is simply the sum of the daily returns for all trading days in that year. Asset returns are usually fat-tailed (leptokurtic), and this is true whether they are measured as log-returns or simple returns. However, log-returns cannot be aggregated across a portfolio to get a portfolio return. This would be possible with simple returns but does not work for log-returns because taking the log is a non-linear transformation process. Therefore the sum of a log is not the same as the sum of a log. In order to calculate portfolio log-returns, it is necessary to calculate the value of the whole portfolio first at each point in time and then to take the log of the portfolio price changes. Your answer has been saved.

Which of the following are alternative names for the dependent variable (usually denoted by y) in linear regression analysis?

(i) The regressand

(ii) The regressor

(iii) The explained variable

(iv) The explanatory variable

	a)	(ii) and (iv) only
	b)	(i) and (iii) only
	c)	(i), (ii), and (iii) only
	d)	(i), (ii), (iii), and (iv)
Correct! Since regressand and explained variable are alternative names for the variable whose movements we are trying to explain. The regressor or explanatory variable are names for x, the variable that is doing the explaining in the model.Incorrect! The correct answer is b, since regressand and explained variable are alternative names for the variable whose movements we are trying to explain. The regressor or explanatory variable are names for x, the variable that is doing the explaining in the model.Your answer has been saved.

Which of the following are alternative names for the independent variable (usually denoted by x) in linear regression analysis?

(i) The regressor

(ii) The regressand

(iii) The causal variable

(iv) The effect variable

	a)	(ii) and (iv) only
	b)	(i) and (iii) only
	c)	(i), (ii), and (iii) only
	d)	(i), (ii), (iii), and (iv)
Correct! The independent variable, usually denoted by x, is also known as the regressor or the causal variable. The regressand and effect variable are alternative names for y. Incorrect! The independent variable, usually denoted by x, is also known as the regressor or the causal variable. The regressand and effect variable are alternative names for y. Your answer has been saved.

Which of the following statements is TRUE concerning the standard regression model?

	a)	y has a probability distribution
	b)	x has a probability distribution
	c)	The disturbance term is assumed to be correlated with x
	d)	For an adequate model, the residual (u-hat) will be zero for all sample data points
Correct! Since y depends on u as well as x, and since u is a random variable, y will also be a random variable. x is assumed to be non-stochastic, i.e. to be fixed and it is therefore not a random variable. Since x is assumed to be non-stochastic, it cannot be correlated with a random variable u, otherwise it would be stochastic! A good model would be one where the residuals are as close to zero as possible. However, unless there is a perfect relationship between y and x (i.e. all of the points lie on a straight line), the residuals cannot all be zero. Incorrect! Since y depends on u as well as x, and since u is a random variable, y will also be a random variable. x is assumed to be non-stochastic, i.e. to be fixed and it is therefore not a random variable. Since x is assumed to be non-stochastic, it cannot be correlated with a random variable u, otherwise it would be stochastic! A good model would be one where the residuals are as close to zero as possible. However, unless there is a perfect relationship between y and x (i.e. all of the points lie on a straight line), the residuals cannot all be zero. Your answer has been saved.

Which of the following statements is TRUE concerning OLS estimation?

	a)	OLS minimises the sum of the vertical distances from the points to the line
	b)	OLS minimises the sum of the squares of the vertical distances from the points to the line
	c)	OLS minimises the sum of the horizontal distances from the points to the line
	d)	OLS minimises the sum of the squares of the horizontal distances from the points to the line.
Correct! OLS minimises the sum of the squares of the vertical distances from the points to the line. The reason that vertical rather than horizontal distances are chosen is due to the set up of the classical linear regression model that assumes x is non-stochastic. Therefore, the question becomes one of how to find the best fitting values of y given the values of x. If we took horizontal distances, this would mean that we were choosing fitted values for x, which wouldn't make sense since x is fixed. The reason that squares of the vertical distances are taken rather than the vertical distances themselves is that some of the points will lie above the fitted line and some below, cancelling each other out. Therefore, a criterion that minimised the sum of the distances would not give unique parameter estimates since an infinite number of lines would satisfy this.Incorrect! OLS minimises the sum of the squares of the vertical distances from the points to the line. The reason that vertical rather than horizontal distances are chosen is due to the set up of the classical linear regression model that assumes x is non-stochastic. Therefore, the question becomes one of how to find the best fitting values of y given the values of x. If we took horizontal distances, this would mean that we were choosing fitted values for x, which wouldn't make sense since x is fixed. The reason that squares of the vertical distances are taken rather than the vertical distances themselves is that some of the points will lie above the fitted line and some below, cancelling each other out. Therefore, a criterion that minimised the sum of the distances would not give unique parameter estimates since an infinite number of lines would satisfy this.Your answer has been saved.

The residual from a standard regression model is defined as

	a)	The difference between the actual value, y, and the mean, y-bar
	b)	The difference between the fitted value, y-hat, and the mean, y-bar
	c)	The difference between the actual value, y, and the fitted value, y-hat
	d)	The square of the difference between the fitted value, y-hat, and the mean, y-bar
Correct! The residual is defined as the difference between the actual value y and the fitted value, y-hat.Incorrect! The residual is defined as the difference between the actual value y and the fitted value, y-hat.Your answer has been saved.

Which one of the following statements best describes the algebraic representation of the fitted regression line?

	a)
	b)
	c)
	d)
Correct! The fitted value for y is obtained by taking the value of the explanatory variable for a particular observation, multiplying it by the slope estimate and adding the intercept estimate. This then gives a value for y-hat from the fitted line for that observation. The answers for a and c are not plausible equations for anything, since the fitted value from the regression model cannot include either a residual or a disturbance in its calculation. The equation in d is a valid equation that splits the actual value y into a part that is explained by the model and a part which the model cannot explain (the residual). However, the equation in d is not the equation for the fitted value.Incorrect! The fitted value for y is obtained by taking the value of the explanatory variable for a particular observation, multiplying it by the slope estimate and adding the intercept estimate. This then gives a value for y-hat from the fitted line for that observation. The answers for a and c are not plausible equations for anything, since the fitted value from the regression model cannot include either a residual or a disturbance in its calculation. The equation in d is a valid equation that splits the actual value y into a part that is explained by the model and a part which the model cannot explain (the residual). However, the equation in d is not the equation for the fitted value.Your answer has been saved.

Which of the following statements concerning the regression population and sample is FALSE?

	a)	The population is the total collection of all items of interest
	b)	The population can be infinite
	c)	In theory, the sample could be larger than the population
	d)	A random sample is one where each individual item from the population is equally likely to be drawn.
Correct! By definition, the population is indeed the collection of all items of interest, and this can be either infinite or finite depending on the context. Also by definition, a random sample is one where each item from the population is equally likely to be drawn. It is of course impossible for the sample to be larger than the population, since the sample takes just some items from the population.Incorrect! By definition, the population is indeed the collection of all items of interest, and this can be either infinite or finite depending on the context. Also by definition, a random sample is one where each item from the population is equally likely to be drawn. It is of course impossible for the sample to be larger than the population, since the sample takes just some items from the population.Your answer has been saved.

Which of the following statements is true concerning the population regression function (PRF) and sample regression function (SRF)?

	a)	The PRF is the estimated model
	b)	The PRF is used to infer likely values of the SRF
	c)	Whether the model is good can be determined by comparing the SRF and the PRF
	d)	The PRF is a description of the process thought to be generating the data.
Correct! The PRF is the true population model for the relationship between the variables x and y. Some researchers draw a distinction between the PRF and data generating process, but the two terms have been used synonymously on this course. The sample is used to estimate a SRF, which is used to determine what are the likely values of the population parameters described by the PRF. Therefore a, b, and c are false and d is a true statement.Incorrect! The PRF is the true population model for the relationship between the variables x and y. Some researchers draw a distinction between the PRF and data generating process, but the two terms have been used synonymously on this course. The sample is used to estimate a SRF, which is used to determine what are the likely values of the population parameters described by the PRF. Therefore a, b, and c are false and d is a true statement.Your answer has been saved.

Which of the following models can be estimated using OLS, following suitable transformations if necessary? (Note that "e" denotes the exponential).

ii)

iii)

iv)

	a)	(i) only
	b)	(i) and (iii) only
	c)	(i), (iii), and (iv) only
	d)	(i), (ii), (iii), and (iv)
Correct! In fact, all of models (i) to (iv) can be estimated using OLS, following suitable transformations where necessary. Clearly (i) is simply the standard model. For (ii), creating a new variable (call it z) as z = e^x, would give the standard model as a regression of y on a constant and z. In (iii), substituting Y = ln(y) and X = ln(x) and regressing Y on a constant and X would again give the standard model. Finally, to estimate (iv), set z = x^2, and regress y on a constant and z. Incorrect! In fact, all of models (i) to (iv) can be estimated using OLS, following suitable transformations where necessary. Clearly (i) is simply the standard model. For (ii), creating a new variable (call it z) as z = e^x, would give the standard model as a regression of y on a constant and z. In (iii), substituting Y = ln(y) and X = ln(x) and regressing Y on a constant and X would again give the standard model. Finally, to estimate (iv), set z = x^2, and regress y on a constant and z. Your answer has been saved.

Which of the following is an equivalent expression for saying that the explanatory variable is "non-stochastic"?

	a)	The explanatory variable is partly random
	b)	The explanatory variable is fixed in repeated samples
	c)	The explanatory variable is correlated with the errors
	d)	The explanatory variable always has a value of one
Correct! The word "stochastic" means random, so non-stochastic means non-random! One of the classical linear regression model assumptions is that the explanatory variable x is non-stochastic or fixed in repeated samples. These are approximately equivalent expressions, although the latter is a slightly stronger statement. Note that "fixed in repeated samples" does not mean that its value is always the same (answer d), and also that this prevents x from being partly random (stochastic) or correlated with the errors (also implying that x is stochastic). Incorrect! The word "stochastic" means random, so non-stochastic means non-random! One of the classical linear regression model assumptions is that the explanatory variable x is non-stochastic or fixed in repeated samples. These are approximately equivalent expressions, although the latter is a slightly stronger statement. Note that "fixed in repeated samples" does not mean that its value is always the same (answer d), and also that this prevents x from being partly random (stochastic) or correlated with the errors (also implying that x is stochastic). Your answer has been saved.

If an estimator is said to be consistent, it is implied that

	a)	On average, the estimated coefficient values will equal the true values
	b)	The OLS estimator is unbiased and no other unbiased estimator has a smaller variance
	c)	The estimates will converge upon the true values as the sample size increases
	d)	The coefficient estimates will be as close to their true values as possible for small and large samples.
Correct! By definition, a consistent estimator is one where the sample estimates converge on their true (population) values as the sample size increases. Answer a is the definition for an unbiased estimator, while b is the result that is proved by the Gauss-Markov theorem. Answer d is a slightly different way of stating the unbiasedness property. Incorrect! By definition, a consistent estimator is one where the sample estimates converge on their true (population) values as the sample size increases. Answer a is the definition for an unbiased estimator, while b is the result that is proved by the Gauss-Markov theorem. Answer d is a slightly different way of stating the unbiasedness property. Your answer has been saved.

If an estimator is said to have minimum variance, which of the following statements is NOT implied?

	a)	The probability that the estimate is a long way away from its true value is minimised
	b)	The estimator is efficient
	c)	Such an estimator would be termed "best"
	d)	Such an estimator will always be unbiased
Correct! An estimator that has minimum variance would also be defined as efficient and "best" - these terms are equivalent to one another. A minimum variance estimator means that the sampling variation in the parameter estimates between one sample and another will be minimised. This is also equivalent to stating that the probability that the estimate for any given sample is a long way off from its true value will be minimised. An estimator can have minimum variance but be a biased estimator. Typically there is an implicit trade off between choosing an unbiased but inefficient estimator and choosing an estimator with a smaller variance that is biased. Incorrect! An estimator that has minimum variance would also be defined as efficient and "best" - these terms are equivalent to one another. A minimum variance estimator means that the sampling variation in the parameter estimates between one sample and another will be minimised. This is also equivalent to stating that the probability that the estimate for any given sample is a long way off from its true value will be minimised. An estimator can have minimum variance but be a biased estimator. Typically there is an implicit trade off between choosing an unbiased but inefficient estimator and choosing an estimator with a smaller variance that is biased. Your answer has been saved.

Consider the OLS estimator for the standard error of the slope coefficient. Which of the following statement(s) is (are) true?

(i) The standard error will be positively related to the residual variance

(ii) The standard error will be negatively related to the dispersion of the observations on the explanatory variable about their mean value

(iii) The standard error will be negatively related to the sample size

(iv) The standard error gives a measure of the precision of the coefficient estimate.

	a)	(ii) and (iv) only
	b)	(i) and (iii) only
	c)	(i), (ii), and (iii) only
	d)	(i), (ii), (iii), and (iv)
Correct! All of statements (i) to (iv) are true. The bigger the residual variance is, the bigger must be the RSS, and therefore the further away are the points from the line. Therefore, the bigger the residual variance is, the bigger will be the coefficient standard errors. This can bee seen since the term "s" appears positively in the standard error formulae for the intercept and the slope. The more dispersed are the observations on the explanatory variable (x) about its mean value, the more precisely the coefficient estimates can be calculated since we would have information about the relationship between y and x over a wider range of values for x. In the formulae, the variation of x about its mean value enters into the denominator for both the slope and the intercept standard errors, so the bigger the dispersion is, the smaller will be the standard errors. The bigger the sample size, the more pieces of information are available from which to estimate the model parameters. The number of observations appears explicitly in the formula for the intercept standard error and implicitly in the formula for the slope standard error. In the latter case, the standard error is inversely related to the sample size since the sum of the squares of the observations on x about their mean value appears in the denominator, and the larger the sample size is, the more terms will be included in this sum. Incorrect! All of statements (i) to (iv) are true. The bigger the residual variance is, the bigger must be the RSS, and therefore the further away are the points from the line. Therefore, the bigger the residual variance is, the bigger will be the coefficient standard errors. This can bee seen since the term "s" appears positively in the standard error formulae for the intercept and the slope. The more dispersed are the observations on the explanatory variable (x) about its mean value, the more precisely the coefficient estimates can be calculated since we would have information about the relationship between y and x over a wider range of values for x. In the formulae, the variation of x about its mean value enters into the denominator for both the slope and the intercept standard errors, so the bigger the dispersion is, the smaller will be the standard errors. The bigger the sample size, the more pieces of information are available from which to estimate the model parameters. The number of observations appears explicitly in the formula for the intercept standard error and implicitly in the formula for the slope standard error. In the latter case, the standard error is inversely related to the sample size since the sum of the squares of the observations on x about their mean value appears in the denominator, and the larger the sample size is, the more terms will be included in this sum. Your answer has been saved.

Which of the following statements is INCORRECT concerning the classical hypothesis testing framework?

	a)	If the null hypothesis is rejected, the alternative is accepted
	b)	The null hypothesis is the statement being tested while the alternative encompasses the remaining outcomes of interest
	c)	The test of significance and confidence interval approaches will always give the same conclusions
	d)	Hypothesis tests are used to make inferences about the population parameters.
Correct! Hypothesis tests are used to make statements about the plausibility of certain values for the population parameters given the estimates made from the sample. By definition, the null hypothesis is the statement being tested while the alternative encompasses other outcomes of interest. The test of significance and confidence interval approaches will always give the same answer (so long as a fixed significance level is used for both) since one can be viewed as just a rearrangement of the other. It is never said that the alternative hypothesis is accepted. The reason that this is not done is that, in general terms, it is possible to reject a null hypothesis without the alternative hypothesis being correct. Therefore a is the only incorrect statement. Incorrect! Hypothesis tests are used to make statements about the plausibility of certain values for the population parameters given the estimates made from the sample. By definition, the null hypothesis is the statement being tested while the alternative encompasses other outcomes of interest. The test of significance and confidence interval approaches will always give the same answer (so long as a fixed significance level is used for both) since one can be viewed as just a rearrangement of the other. It is never said that the alternative hypothesis is accepted. The reason that this is not done is that, in general terms, it is possible to reject a null hypothesis without the alternative hypothesis being correct. Therefore a is the only incorrect statement. Your answer has been saved.

Suppose that a hypothesis test is conducted using a 5% significance level. Which of the following statements are correct?

(i) The significance level is equal to the size of the test

(ii) The significance level is equal to the power of the test

(iii) 2.5% of the total distribution will be in each tail rejection region for a 2-sided test

(iv) 5% of the total distribution will be in each tail rejection region for a 2-sided test.

	a)	(ii) and (iv) only
	b)	(i) and (iii) only
	c)	(i), (ii), and (iii) only
	d)	(i), (ii), (iii), and (iv)
Correct! (i) and (iii) are true while (ii) and (iv) are false. The significance level and the size of the test are different ways of saying the same thing: they measure the proportion of the total distribution of that the test statistic is assumed to follow which is placed in the rejection region(s). The significance level is equal to the probability of type I error. The probability of type II error is given by the power of the test, not the significance level. If a 5% significance level is used, this implies that in total 5% of the whole distribution must be in the rejection region and if the test is 2-sided, this means that 2.5% of the distribution will be in each of the rejection regions, and not 5% in each tail. 5% in each tail would imply a 10% significance level for a 2-sided test, while the question specifies a 5% significance level.Incorrect! (i) and (iii) are true while (ii) and (iv) are false. The significance level and the size of the test are different ways of saying the same thing: they measure the proportion of the total distribution of that the test statistic is assumed to follow which is placed in the rejection region(s). The significance level is equal to the probability of type I error. The probability of type II error is given by the power of the test, not the significance level. If a 5% significance level is used, this implies that in total 5% of the whole distribution must be in the rejection region and if the test is 2-sided, this means that 2.5% of the distribution will be in each of the rejection regions, and not 5% in each tail. 5% in each tail would imply a 10% significance level for a 2-sided test, while the question specifies a 5% significance level.Your answer has been saved.

Consider an identical situation to that of question 21, except that now a 2-sided alternative is used. What would now be the appropriate conclusion?

	a)	H0 is rejected
	b)	H0 is not rejected
	c)	H1 is rejected
	d)	There is insufficient information given in the question to reach a conclusion
Correct! Now, if a 2-sided test is used, the test statistic would still take the same value, and rejection would occur if the test statistic fell in either region. Since the 5% 2-sided critical values are close to -2 and +2, the statistic is clearly now in the rejection region, and hence a is correct.Incorrect! Now, if a 2-sided test is used, the test statistic would still take the same value, and rejection would occur if the test statistic fell in either region. Since the 5% 2-sided critical values are close to -2 and +2, the statistic is clearly now in the rejection region, and hence a is correct.Your answer has been saved.

Which one of the following would be the most appropriate as a 95% (two-sided) confidence interval for the intercept term of the model given in question 21?

	a)	(-4.79,2.19)
	b)	(-4.16,4.16)
	c)	(-1.98,1.98)
	d)	(-5.46,2.86)
Correct! Recall that the formula for estimating a confidence interval for the intercept parameter would be (alphahat - SE(alphahat)Xcritical_value, alphahat + SE(alphahat)Xcritical_value ) putting the relevant terms would give the interval in this case as (-1.3-1.98X2.1, -1.3+1.98X2.1) or (-5.46,2.86). Therefore d is the correct answer. Errors that you could have made would include using the one-sided 5% critical value, which would be about 1.66 instead of 1.98. This would have given answer a. The second possible error would be to forget to add in the coefficient value, so that the interval would be wrongly calculated as (-1.98X2.1, 1.98X2.1), which would give answer b. Answer c would have been obtained if the critical values alone had been used! Incorrect! Recall that the formula for estimating a confidence interval for the intercept parameter would be (alphahat - SE(alphahat)Xcritical_value, alphahat + SE(alphahat)Xcritical_value ) putting the relevant terms would give the interval in this case as (-1.3-1.98X2.1, -1.3+1.98X2.1) or (-5.46,2.86). Therefore d is the correct answer. Errors that you could have made would include using the one-sided 5% critical value, which would be about 1.66 instead of 1.98. This would have given answer a. The second possible error would be to forget to add in the coefficient value, so that the interval would be wrongly calculated as (-1.98X2.1, 1.98X2.1), which would give answer b. Answer c would have been obtained if the critical values alone had been used! Your answer has been saved.

Which one of the following is the most appropriate definition of a 99% confidence interval?

	a)	99% of the time in repeated samples, the interval would contain the true value of the parameter
	b)	99% of the time in repeated samples, the interval would contain the estimated value of the parameter
	c)	99% of the time in repeated samples, the null hypothesis will be rejected
	d)	99% of the time in repeated samples, the null hypothesis will not be rejected when it was false
Correct! Although from a philosophical perspective, some researchers would disagree with this definition, on this course a 99% confidence interval is taken to mean that 99% of the time in repeated samples, the interval would contain the true parameter value. Thus a is correct. Of course, by construction the interval will always contain the parameter estimate exactly in the middle, so b is incorrect. For a 99% confidence interval, we can say that 99% of the time the null would not be rejected when the null was correct (i.e. we made the right decision), which is not the formulation of d, so d is incorrect. We cannot say how often the null hypothesis will be rejected - it depends on whether it is right or wrong! All we could say is how often the null would be rejected as a result of chance alone. Therefore c is incorrect. Incorrect! Although from a philosophical perspective, some researchers would disagree with this definition, on this course a 99% confidence interval is taken to mean that 99% of the time in repeated samples, the interval would contain the true parameter value. Thus a is correct. Of course, by construction the interval will always contain the parameter estimate exactly in the middle, so b is incorrect. For a 99% confidence interval, we can say that 99% of the time the null would not be rejected when the null was correct (i.e. we made the right decision), which is not the formulation of d, so d is incorrect. We cannot say how often the null hypothesis will be rejected - it depends on whether it is right or wrong! All we could say is how often the null would be rejected as a result of chance alone. Therefore c is incorrect. Your answer has been saved.

Which one of the following statements best describes a Type II error?

	a)	It is the probability of incorrectly rejecting the null hypothesis
	b)	It is equivalent to the power of the test
	c)	It is equivalent to the size of the test
	d)	It is the probability of failing to reject a null hypothesis that was wrong
Correct! By definition, a type II error occurs when failing to reject a null hypothesis that was wrong. Thus d is correct. The situation of incorrectly rejecting the null hypothesis when the null was true is a type I error. This probability of type I error is equal to the size of the test. The power of the test is one minus the probability of type II error, not the probability of type II error itself. Incorrect! By definition, a type II error occurs when failing to reject a null hypothesis that was wrong. Thus d is correct. The situation of incorrectly rejecting the null hypothesis when the null was true is a type I error. This probability of type I error is equal to the size of the test. The power of the test is one minus the probability of type II error, not the probability of type II error itself. Your answer has been saved.

Suppose that a test statistic has associated with it a p-value of 0.08. Which one of the following statements is true?

(i) If the size of the test were exactly 8%, we would be indifferent between rejecting and not rejecting the null hypothesis

(ii) The null would be rejected if a 10% size of test were used

(iii) The null would not be rejected if a 1% size of test were used

(iv) The null would be rejected if a 5% size of test were used.

	a)	(ii) and (iv) only
	b)	(i) and (iii) only
	c)	(i), (ii), and (iii) only
	d)	(i), (ii), (iii), and (iv)
Correct! (i) to (iii) are correct while (iv) is incorrect. The p-value is defined as the marginal significance level where we would be indifferent between rejecting and not rejecting the null hypothesis. This is also sometimes termed the exact significance level, or the plausibility level for the null hypothesis. Thus, under this definition, it is clear that if a significance level (size of test) were exactly 8%, we would be indifferent between rejecting and not rejecting the null hypothesis (so i is true). If we used a larger size of test (i.e. a bigger rejection region), the null would be rejected (so ii is true), while if we used a smaller size of test (a smaller rejection region), the null would not be rejected (so iv is false and iii is true).Incorrect! (i) to (iii) are correct while (iv) is incorrect. The p-value is defined as the marginal significance level where we would be indifferent between rejecting and not rejecting the null hypothesis. This is also sometimes termed the exact significance level, or the plausibility level for the null hypothesis. Thus, under this definition, it is clear that if a significance level (size of test) were exactly 8%, we would be indifferent between rejecting and not rejecting the null hypothesis (so i is true). If we used a larger size of test (i.e. a bigger rejection region), the null would be rejected (so ii is true), while if we used a smaller size of test (a smaller rejection region), the null would not be rejected (so iv is false and iii is true).Your answer has been saved.

Suppose that observations are available on the monthly bond prices of 100 companies for 5 years. What type of data are these?

	a)	Cross-sectional
	b)	Time-series
	c)	Panel
	d)	Qualitative
Correct! Since the data have the dimensions of both time series (5 years of observations) and of cross-sections (100 companies), this would be known as a panel data set. A cross-sectional series would not have data over a period of time, while a time-series data set would use information on one company at a time. Bond prices are clearly an example of quantitative rather than qualitative data, since they can take on any (non-negative) values and are not constrained to take on only certain values as qualitative data would be.Incorrect! Since the data have the dimensions of both time series (5 years of observations) and of cross-sections (100 companies), this would be known as a panel data set. A cross-sectional series would not have data over a period of time, while a time-series data set would use information on one company at a time. Bond prices are clearly an example of quantitative rather than qualitative data, since they can take on any (non-negative) values and are not constrained to take on only certain values as qualitative data would be.Your answer has been saved.

Which of the following is not a discrete probability distribution?

1. Which of these is not a discrete probability distribution? Explanation: Hyper geometric distribution, Binomial distribution, and Poisson distribution are all part of discrete probability distribution family. But, Normal distribution is a Continuous distribution.

Which statement is not true for a probability distribution for a discrete random variable?

Expert-Verified Answer Random variable can only have one value" is not true. A variable in an equation can satisfy the equation such that its value is zero. So, it is possible that the value of a random variable is zero. A random can have a single value but it is not necessary that it only has one value.

What is true about discrete probability distribution?

In a discrete probability distribution, the sum of the probabilities must equal 1, and all probabilities must be greater than or equal to 0 and less than or equal to 1.

Which of the following is true of both discrete and continuous probability distributions?

That the sum of all probabilities is equal to one.

Which of the following statements is not true of a discrete probability distribution?

Which of the following is not a discrete probability distribution?

Which statement is not true for a probability distribution for a discrete random variable?

What is true about discrete probability distribution?

Which of the following is true of both discrete and continuous probability distributions?

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