Question 1

Math SAT scores (Y)are normally distributed with a mean of 500 and a standard deviation of 100. An evening school advertises that it can improve students' scores by roughly a third of a standard deviation, or 30 points, if they attend a course which runs over several weeks. (A similar claim is made for attending a verbal SAT course.)The statistician for a consumer protection agency suspects that the courses are not effective. She views the situation as follows: H₀ : $\mu _ { Y }$ = 500 vs. H₁ : $\mu _ { Y }$ = 530. (a)Sketch the two distributions under the null hypothesis and the alternative hypothesis. (b)The consumer protection agency wants to evaluate this claim by sending 50 students to attend classes. One of the students becomes sick during the course and drops out. What is the distribution of the average score of the remaining 49 students under the null, and under the alternative hypothesis? (c)Assume that after graduating from the course, the 49 participants take the SAT test and score an average of 520. Is this convincing evidence that the school has fallen short of its claim? What is the p-value for such a score under the null hypothesis? (d)What would be the critical value under the null hypothesis if the size of your test were 5%? (e)Given this critical value, what is the power of the test? What options does the statistician have for increasing the power in this situation?

Accepted Answer

The answer of Math SAT scores (Y)are normally distributed with...

Question 2

The following statement about the sample correlation coefficient is true. A)-1 ? r_X,Y ? 1. B) ${ r} ^ { 2 } _{ X Y }$ $\stackrel { p } { \rightarrow }$ corr(X_i, Y_i). C)| r_X,Y |< 1. D)r_X,Y = $\frac { S _ { x y } ^ { 2 } } { S _ { x } ^ { 2 } S _ { y } ^ { 2 } }$

Accepted Answer

This statement accurately reflects the range of possible values for the sample correlation coefficient. The value of r_X,Y can range from -1 to 1, indicating the strength and direction of the relationship between two variables.

Question 3

When the sample size n is large, the 90% confidence interval for ${ }^{\mu} Y$
 is
A) $\bar { Y }$ ± 1.96SE( $\bar { Y }$ ).
B) $\bar { Y }$ ± 1.64SE( $\bar { Y }$ ).
C) $\bar { Y }$ ± 1.64 ${ } ^ { \sigma } Y$
D) $\bar { Y }$ ± 1.96.

Accepted Answer

For a 90% confidence interval, the z-score is approximately 1.64, so the interval is $\bar { Y }$ ± 1.64SE( $\bar { Y }$ ).

Question 4

Degrees of freedom
A)in the context of the sample variance formula means that estimating the mean uses up some of the information in the data.
B)is something that certain undergraduate majors at your university/college other than economics seem to have an ? amount of.
C)are (n-2)when replacing the population mean by the sample mean.
D)ensure that $S _ { y } ^ { 2 } = \sigma _ { y } ^ { 2 }$

Accepted Answer

Estimating the mean from the sample data uses up one degree of freedom, which is why the sample variance formula uses (n-1) instead of n.

Question 5

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.

Accepted Answer

The answer of Think of at least nine examples, three...

Question 6

Assume that two presidential candidates, call them Bush and Gore, receive 50% of the votes in the population. You can model this situation as a Bernoulli trial, where Y is a random variable with success probability Pr(Y = 1)= p, and where Y = 1 if a person votes for Bush and Y = 0 otherwise. Furthermore, let $\hat { p }$ be the fraction of successes (1s)in a sample, which is distributed N(p, $\frac { p ( 1 - p ) } { n }$ )in reasonably large samples, say for n ? 40.
(a)Given your knowledge about the population, find the probability that in a random sample of 40, Bush would receive a share of 40% or less.
(b)How would this situation change with a random sample of 100?
(c)Given your answers in (a)and (b), would you be comfortable to predict what the voting intentions for the entire population are if you did not know p but had polled 10,000 individuals at random and calculated $\hat { p }$ ? Explain.
(d)This result seems to hold whether you poll 10,000 people at random in the Netherlands or the United States, where the former has a population of less than 20 million people, while the United States is 15 times as populous. Why does the population size not come into play?

Accepted Answer

The answer of Assume that two presidential candidates, call them...

Question 7

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.

Accepted Answer

The t-statistic follows a Student's t-distribution with n-1 degrees of freedom when the sample data is normally distributed.

Question 8

The power of the test A)is the probability that the test actually incorrectly rejects the null hypothesis when the null is true. B)depends on whether you use $\bar { Y }$ or $\bar { Y }$ ² for the t-statistic. C)is one minus the size of the test. D)is the probability that the test correctly rejects the null when the alternative is true.

Accepted Answer

The power of the test is the probability that the test correctly rejects the null when the alternative is true. Answer A refers to the significance level or type I error rate, and answer C is the complement of the significance level. Answer B is irrelevant to the definition of power.

Question 9

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) $\frac { 1 } { n - 1 } \sum _ { i = 1 } ^ { n } X _ { i } Y _ { i } - \frac { n } { n - 1 } \overline { X Y }$ C) $\frac { 1 } { n } \sum _ { i = 1 } ^ { n } \left( X _ { i } - \mu _ { X } \right) \left( Y _ { i } - \mu _ { Y } \right)$ D)r_XYS_YS_Y, where r_XY is the correlation coefficient.

Accepted Answer

The sample covariance is calculated using the formula $\frac { 1 } { n - 1 } \sum _ { i = 1 } ^ { n } \left( X _ { i } - \bar { X } \right) \left( Y _ { i } - \bar { Y } \right)$ or $\frac { 1 } { n - 1 } \sum _ { i = 1 } ^ { n } X _ { i } Y _ { i } - \frac { n } { n - 1 } \overline { X Y }$. The formula $\frac { 1 } { n } \sum _ { i = 1 } ^ { n } \left( X _ { i } - \mu _ { X } \right) \left( Y _ { i } - \mu _ { Y } \right)$ is used to calculate the population covariance, not the sample covariance. The formula r_XYS_YS_Y, where r_XY is the correlation coefficient, is also used to calculate the covariance, but it is not one of the options given in the question.

Question 10

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 4^th to 6^th grade as part of her science project. The accompanying table shows her findings. Height of Young Boys and Girls, Grades 4-6, in inches (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. (b)Find the difference in height and the standard error of the difference. (c)Generate a 95% confidence interval for the difference in height. (d)Calculate the t-statistic for comparing the two means. Is the difference statistically significant at the 1% level? Which critical value did you use? Why would this number be smaller if you had assumed a one-sided alternative hypothesis? What is the intuition behind this?

Accepted Answer

The answer of Adult males are taller, on average, than...

Question 11

When testing for differences of means, the t-statistic t = $\frac { \bar { Y } m - \bar { Y } w } { S E ( \bar { Y } m - \overline { \bar { Y } w } ) }$ , where $S E \left[ \bar { Y } _ { m } - \bar { Y } _ { W } \right) = \sqrt { \frac { s _ { m } ^ { 2 } } { n _ { m } } + \frac { s _ { w } ^ { 2 } } { n _ { w } } }$ has A)a student t distribution if the population distribution of Y is not normal B)a student t distribution if the population distribution of Y is normal C)a normal distribution even in small samples D)cannot be computed unless $n _ { w }$ = $n _ { m }$

Accepted Answer

The t-statistic follows a student t distribution if the population distribution of Y is normal.

Question 12

Your packaging company fills various types of flour into bags. Recently there have been complaints from one chain of stores: a customer returned one opened 5 pound bag which weighed significantly less than the label indicated. You view the weight of the bag as a random variable which is normally distributed with a mean of 5 pounds, and, after studying the machine specifications, a standard deviation of 0.05 pounds.
(a)You take a sample of 20 bags and weigh them. Sketch below what the average pattern of individual weights might look like. Let the horizontal axis indicate the sampled bag number (1, 2, …, 20). On the vertical axis, mark the expected value of the weight under the null hypothesis, and two (? 1.96)standard deviations above and below the expected value. Draw a line through the graph for E(Y)+ 2 ${ } ^ { \sigma } { } _ { Y }$ , E(Y), and E(Y)- 2 ${ } ^ { \sigma } Y$ How many of the bags in a sample of 20 will you expect to weigh either less than 4.9 pounds or more than 5.1 pounds?
(b)You sample 25 bags of flour and calculate the average weight. What is the distribution of the average weight of these 25 bags? Repeating the same exercise 20 times, sketch what the distribution of the average weights would look like in a graph similar to the one you drew in (b), where you have adjusted the standard error of $\bar { Y }$ accordingly.
(c)For each of the twenty observations in (c)a 95% confidence interval is constructed. Draw these confidence intervals, using the same graph as in (c). How many of these 20 confidence intervals would you expect to weigh 5 pounds under the null hypothesis?

Accepted Answer

The answer of Your packaging company fills various types of...

Question 13

You have collected data on the average weekly amount of studying time (T)and grades (G)from the peers at your college. Changing the measurement from minutes into hours has the following effect on the correlation coefficient:
A)decreases the $r _ { T G }$ by dividing the original correlation coefficient by 60
B)results in a higher $r _ { T G }$
C)cannot be computed since some students study less than an hour per week
D)does not change the $r _ { T G }$

Accepted Answer

The correlation coefficient is a unitless measure of the strength and direction of a linear relationship between two variables. Changing the units of measurement does not affect the correlation coefficient.

Question 14

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.

Accepted Answer

The correlation coefficient is a statistical measure of linear association between two variables, which means it measures the strength and direction of the linear relationship between them. Therefore, option B is correct. Option A is partially correct, as the correlation coefficient always lies between -1 and 1 (not zero and one). Option C is incorrect, as correlation does not imply causation. Option D is incorrect, as the correlation coefficient measures the strength of the linear relationship, not nonlinear relationships.

Question 15

The standard error for the difference in means if two random variables M and W, when the two population variances are different, is
A) $\sqrt { \frac { S _ { M} ^ { 2 } + S _ { W } ^ { 2 } } { n _ {M} + n _ { W } } }$
B) $\frac { S _ { M} } { n _ {W } } + \frac { S _ { M } } { n _ { W } }$
C) $\sqrt { \sqrt { \frac { 1 } { 2 } \left( \frac { S _ { M } ^ { 2 } + S _ { W} ^ { 2 } } { n _ { M } + n _ {W} } \right) } }$
D) $\sqrt { \frac { S _ {M} ^ { 2 } + S _ { W} ^ { 2 } } { n _ { M } + n _ { W} } }$

Accepted Answer

The standard error for the difference in means if two random variables M and W, when the two population variances are different, is $\sqrt { \frac { S _ { M} ^ { 2 } + S _ { W} ^ { 2 } } { n _ { M } + n _ { W} } }$

Question 16

The t-statistic is defined as follows:
A) $t = \frac { \frac { \bar { Y } - \mu _ { y , \mathrm { D } } } { \sigma _ { y } ^ { 2 } } } { n }$
B) $t = \frac { \bar { Y } - \mu _ { \gamma , \mathrm { D } } } { S E [ \bar { Y } ) }$
C) $t = \frac { \left( \bar { Y } - \mu _ { \gamma , 0 } \right) ^ { 2 } } { S E [ \bar { Y } ) }$
D)1.96.

Accepted Answer

The answer of The t-statistic is defined as follows:
A) \(t...

Question 17

When testing for differences of means, you can base statistical inference on the
A)Student t distribution in general
B)normal distribution regardless of sample size
C)Student t distribution if the underlying population distribution of Y is normal, the two groups have the same variances, and you use the pooled standard error formula
D)Chi-squared distribution with ( $n _ { w }$ + $n _ { m }$ - 2)degrees of freedom

Accepted Answer

The Student t distribution can be used when the underlying population distribution of Y is normal, the two groups have the same variances, and the pooled standard error formula is used. This is the most appropriate approach for testing differences of means. The normal distribution can only be used when the sample size is large (usually n > 30). The chi-squared distribution is used for testing independence between two categorical variables.

Question 18

A low correlation coefficient implies that
A)the line always has a flat slope
B)in the scatterplot, the points fall quite far away from the line
C)the two variables are unrelated
D)you should use a tighter scale of the vertical and horizontal axis to bring the observations closer to the line

Accepted Answer

A low correlation coefficient indicates that in a scatterplot, the points do not closely follow a straight line, meaning they are spread out or fall quite far from the line of best fit. This does not necessarily mean the variables are unrelated (C) or that adjustments to the scale (D) or the slope of the line (A) would correct the interpretation of the correlation.

Question 19

Assume that you have 125 observations on the height (H)and weight (W)of your peers in college. Let ${ } ^ { S } H W$ = 68, ${ } ^ { S } H$ = 3.5, $\text { SW }$ = 29. The sample correlation coefficient is
A)1.22
B)0.50
C)0.67
D)Cannot be computed since males and females have not been separated out.

Accepted Answer

The sample correlation coefficient is a measure of the linear association between two variables, and it ranges between -1 and 1. A value of 1 indicates a perfect positive correlation, 0 indicates no correlation, and -1 indicates a perfect negative correlation. The sample correlation coefficient between height and weight in this scenario is 0.67, which indicates a moderate positive correlation. Choice D is incorrect because whether the individuals are male or female is not relevant to calculating the correlation between height and weight.

Question 20

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) $S _ { y } ^ { 2 } = \frac { 1 } { n - 1 } \sum _ { i = 1 } ^ { n - 1 } \left( Y _ {i } - \bar { Y } \right) ^ { 2 }$

Accepted Answer

The sample variance formula is $S _ { y } ^ { 2 } = \frac { 1 } { n - 1 } \sum _ { i = 1 } ^ { n } \left( Y _ { i} - \bar { Y } \right) ^ { 2 }$, which calculates the average of the squared deviations from the sample mean.

Quiz 3: Review of Statistics

The following statement about the sample correlation coefficient is true.

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

Degrees of freedom

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.

The t-statistic has the following distribution:

The power of the test

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

You have collected data on the average weekly amount of studying time (T) and grades (G) from the peers at your college. Changing the measurement from minutes into hours has the following effect on the correlation coefficient:

The correlation coefficient

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

The t-statistic is defined as follows:

When testing for differences of means, you can base statistical inference on the

A low correlation coefficient implies that

Assume that you have 125 observations on the height (H) and weight (W) of your peers in college. Let ${ } ^ { S } H W$ = 68, ${ } ^ { S } H$ = 3.5, $\text { SW }$ = 29. The sample correlation coefficient is

The formula for the sample variance is

Quiz 3: Review of Statistics

The following statement about the sample correlation coefficient is true.

When the sample size n is large, the 90% confidence interval for μY{ }^{\mu} YμY is

Degrees of freedom

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.

The t-statistic has the following distribution:

The power of the test

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

You have collected data on the average weekly amount of studying time (T) and grades (G) from the peers at your college. Changing the measurement from minutes into hours has the following effect on the correlation coefficient:

The correlation coefficient

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

The t-statistic is defined as follows:

When testing for differences of means, you can base statistical inference on the

A low correlation coefficient implies that

Assume that you have 125 observations on the height (H) and weight (W) of your peers in college. Let SHW{ } ^ { S } H WSHW = 68, SH{ } ^ { S } HSH = 3.5, SW \text { SW } SW = 29. The sample correlation coefficient is

The formula for the sample variance is

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

Assume that you have 125 observations on the height (H) and weight (W) of your peers in college. Let ${ } ^ { S } H W$ = 68, ${ } ^ { S } H$ = 3.5, $\text { SW }$ = 29. The sample correlation coefficient is