MATH 225N WEEK 5 FINAL PAPER 2026
COMPLETE QUESTIONS AND SOLUTIONS
◉ A variable that is related to either the response variable or the
predictor variable or both, but which is excluded from the analysis is
a. Answer: lurking variable
◉ The coefficient of determination is the __________ of the linear
correlation coefficient.. Answer: square
◉ Provide an appropriate response.
Civil engineers often use the straight-line equation, E(y) = β0 + β1x,
to model the relationship between the mean shear strength E(y) of
masonry joints and precompression stress, x. To test this theory, a
series of stress tests were performed on solid bricks arranged in
triplets and joined with mortar. The precompression stress was
varied for each triplet and the ultimate shear load just before failure
(called the shear strength) was recorded. The stress results for n = 7
triplet tests is shown in the accompanying table followed by a SAS
printout of the regression analysis.
For this data set, R2 = .9052.
,Give a practical interpretation of R2, the coefficient of determination
for the least squares model. Express R2 to the nearest whole
percent.. Answer: About 91% of the total variation in the sample of
y-values can be explained by (or attributed to) the linear
relationship between shear strength and precompression stress.
◉ Each year a nationally recognized publication conducts its "Survey
of America's Best Graduate and Professional Schools." An academic
advisor wants to predict the typical starting salary of a graduate at a
top business school using GMAT score of the school as a predictor
variable. Total GMAT scores range from 200 to 800. A simple linear
regression of SALARY versus GMAT using 25 data points yields the
regression equation given below:
y = 228x - 92,040
Give an interpretation of the y-intercept.. Answer: The value has no
practical interpretation since a GMAT of 0 is nonsensical and outside
the range of the sample data.
◉ Find the indicated probability. If necessary, round to three decimal
places.
,Suppose that E and F are two events and that N(E and F) = 450 and
N(E) = 870. What is P(E)?. Answer: 0.517
◉ In a contest in which 7 contestants are entered, in how many ways
can the 4 distinct prizes be awarded?. Answer: 840
◉ Solve the problem.
In the game of craps two dice are rolled and the up faces are totaled.
If the person rolling the dice on the first roll rolls a 7 or an 11 total
they win. If they roll a 2, 3, or 12 on the first roll they lose. If they roll
any other total then on subsequent rolls they must roll that total
before rolling a 7 to win. What is the probability of winning on the
first roll?. Answer: 0.22
◉ Suppose a basketball player is an excellent free throw shooter and
makes 91% of his free throws (i.e., he has a 91% chance of making a
single free throw). Assume that free throw shots are independent of
one another. Suppose this player gets to shoot four free throws. Find
the probability that he misses all four consecutive free throws.
Round to the nearest ten-thousandth.. Answer: 0.0001
◉ Provide an appropriate response.
, A single die is rolled twice. The set of 36 equally likely outcomes is
{(1, 1), (1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (2, 5), (2, 6), (3, 1), (3, 2), (3,
3), (3, 4), (3, 5), (3, 6), (4, 1), (4, 2), (4, 3), (5, 3), (5, 4), (5, 5), (5, 6),
(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}. Find the probability of
getting two numbers whose sum is less than 13.. Answer: 1
◉ The complement of 4 heads in the toss of 4 coins is. Answer: at
least one tail.
◉ How many distinct arrangements can be formed from all the
letters of "students"?. Answer: 10,080
◉ A human gene carries a certain disease from the mother to the
child with a probability rate of 57%. That is, there is a 57% chance
that the child becomes infected with the disease. Suppose a female
carrier of the gene has four children. Assume that the infections of
the four children are independent of one another. Find the
probability that at least one of the children get the disease from
their mother. Round to the nearest thousandth.. Answer: 0.966
◉ In a carnival game, a person wagers $2 on the roll of two dice. If
the total of the two dice is 2, 3, 4, 5, or 6 then the person gets $4 (the
$2 wager and $2 winnings). If the total of the two dice is 8, 9, 10, 11,
or 12 then the person gets nothing (loses $2). If the total of the two
dice is 7, the person gets $0.75 back (loses $0.25). What is the
expected value of playing the game once?. Answer: $-0.04
COMPLETE QUESTIONS AND SOLUTIONS
◉ A variable that is related to either the response variable or the
predictor variable or both, but which is excluded from the analysis is
a. Answer: lurking variable
◉ The coefficient of determination is the __________ of the linear
correlation coefficient.. Answer: square
◉ Provide an appropriate response.
Civil engineers often use the straight-line equation, E(y) = β0 + β1x,
to model the relationship between the mean shear strength E(y) of
masonry joints and precompression stress, x. To test this theory, a
series of stress tests were performed on solid bricks arranged in
triplets and joined with mortar. The precompression stress was
varied for each triplet and the ultimate shear load just before failure
(called the shear strength) was recorded. The stress results for n = 7
triplet tests is shown in the accompanying table followed by a SAS
printout of the regression analysis.
For this data set, R2 = .9052.
,Give a practical interpretation of R2, the coefficient of determination
for the least squares model. Express R2 to the nearest whole
percent.. Answer: About 91% of the total variation in the sample of
y-values can be explained by (or attributed to) the linear
relationship between shear strength and precompression stress.
◉ Each year a nationally recognized publication conducts its "Survey
of America's Best Graduate and Professional Schools." An academic
advisor wants to predict the typical starting salary of a graduate at a
top business school using GMAT score of the school as a predictor
variable. Total GMAT scores range from 200 to 800. A simple linear
regression of SALARY versus GMAT using 25 data points yields the
regression equation given below:
y = 228x - 92,040
Give an interpretation of the y-intercept.. Answer: The value has no
practical interpretation since a GMAT of 0 is nonsensical and outside
the range of the sample data.
◉ Find the indicated probability. If necessary, round to three decimal
places.
,Suppose that E and F are two events and that N(E and F) = 450 and
N(E) = 870. What is P(E)?. Answer: 0.517
◉ In a contest in which 7 contestants are entered, in how many ways
can the 4 distinct prizes be awarded?. Answer: 840
◉ Solve the problem.
In the game of craps two dice are rolled and the up faces are totaled.
If the person rolling the dice on the first roll rolls a 7 or an 11 total
they win. If they roll a 2, 3, or 12 on the first roll they lose. If they roll
any other total then on subsequent rolls they must roll that total
before rolling a 7 to win. What is the probability of winning on the
first roll?. Answer: 0.22
◉ Suppose a basketball player is an excellent free throw shooter and
makes 91% of his free throws (i.e., he has a 91% chance of making a
single free throw). Assume that free throw shots are independent of
one another. Suppose this player gets to shoot four free throws. Find
the probability that he misses all four consecutive free throws.
Round to the nearest ten-thousandth.. Answer: 0.0001
◉ Provide an appropriate response.
, A single die is rolled twice. The set of 36 equally likely outcomes is
{(1, 1), (1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (2, 5), (2, 6), (3, 1), (3, 2), (3,
3), (3, 4), (3, 5), (3, 6), (4, 1), (4, 2), (4, 3), (5, 3), (5, 4), (5, 5), (5, 6),
(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}. Find the probability of
getting two numbers whose sum is less than 13.. Answer: 1
◉ The complement of 4 heads in the toss of 4 coins is. Answer: at
least one tail.
◉ How many distinct arrangements can be formed from all the
letters of "students"?. Answer: 10,080
◉ A human gene carries a certain disease from the mother to the
child with a probability rate of 57%. That is, there is a 57% chance
that the child becomes infected with the disease. Suppose a female
carrier of the gene has four children. Assume that the infections of
the four children are independent of one another. Find the
probability that at least one of the children get the disease from
their mother. Round to the nearest thousandth.. Answer: 0.966
◉ In a carnival game, a person wagers $2 on the roll of two dice. If
the total of the two dice is 2, 3, 4, 5, or 6 then the person gets $4 (the
$2 wager and $2 winnings). If the total of the two dice is 8, 9, 10, 11,
or 12 then the person gets nothing (loses $2). If the total of the two
dice is 7, the person gets $0.75 back (loses $0.25). What is the
expected value of playing the game once?. Answer: $-0.04
