Aldrich Ames is a convicted traitor who leaked American secrets to a foreign power. Yet Ames took routine lie detector tests and each time passed them. How can this be done? Recognizing control questions, employing unusual breathing patterns, biting one's tongue at the right time, pressing one's toes hard to the floor, and counting backwards by 7 are countermeasures that are difficult to detect but can change the results of a polygraph examination†. In fact, it is reported in Professor Ford's book that after only 20 minutes of instruction by "Buzz" Fay (a prison inmate), 85% of those trained were able to pass the polygraph examination even when guilty of a crime. Suppose that a random sample of five students (in a psychology laboratory) are told a "secret" and then given instructions on how to pass the polygraph examination without revealing their knowledge of the secret. What are the following probabilities? (Round your answers to three decimal places.)

(a) all the students are able to pass the polygraph examination


(b) more than half the students are able to pass the polygraph examination


(c) no more than half of the students are able to pass the polygraph examination


(d) all the students fail the polygraph examination

Suppose approximately 80% of all marketing personnel are extroverts, whereas about 65% of all computer programmers are introverts. (Round your answers to three decimal places.)

(a) At a meeting of 15 marketing personnel, what is the probability that 10 or more are extroverts?


What is the probability that 5 or more are extroverts?


What is the probability that all are extroverts?


(b) In a group of 4 computer programmers, what is the probability that none are introverts?


What is the probability that 2 or more are introverts?


What is the probability that all are introverts?

1. Confidence interval for the difference between the two population means.
(Assume that the two samples are independent simple random samples selected from normally distributed populations.)

A researcher was interested in comparing the GPAs of students at two different colleges. Independent simple random samples of 8 students from college A and 13 students from college B yielded the following summary statistics:

  
Construct a 95% confidence interval for μ₁ – μ₂, the difference between the mean GPA of students in college A and the mean GPA of students in college B .

Select one:

A.-0.78 < μ₁ – μ₂< 0.13

B, -0.84 < μ₁ – μ₂< 0.19

C, -0.80 < μ₁ – μ₂< 0.15

D, -0.75 < μ₁ – μ₂< 0.18

2.

A researcher was interested in comparing the response times of two different cab companies. Companies A and B were each called at n = 36 randomly selected times. The calls to company A were made independently of the calls to company B. The response times were recorded and the summary statistics were as follows:

Find the margin of error, E, for a 98% confidence interval that can be used to estimate the difference between the mean resting pulse rate of people who do not exercise regularly and the mean resting pulse rate of people who do. Round your answer to two decimal places.
(Note: Use Table A-3 for the critical value needed in the formula)

(b) Professor GeniusAtCalculus has two lecture sections (A and B) of the same 4th year
Advanced Calculus (AMA 4301) course in Semester 2. She wants to investigate whether
section A students maybe ”smarter” than section B students by comparing their performances in the midterm test. A random sample of 12 students were taken from section
A, with mean midterm test score of 78.8 and standard deviation 8.5; and a random
sample of 9 students were taken from section B, with mean midterm test score of 86
and standard deviation 9.3. Assume the population standard deviations of midterm test
scores for both sections are the same. Construct the 90% confidence interval for the
difference in midterm test scores of the two sections. Based on the sample midterm test
scores from the two sections, can Professor GeniusAtCalculus conclude that there is any
evidence that one section of students are ”smarter” than the other section? Justify your
conclusions.
[8 marks]

(c) The COVID-19 (coronavirus) mortality rate of a you country is defined as the ratio of the
number of deaths due to COVID-19 divided by the number of (confirmed) cases of
COVID-19 in that country. Suppose we want to investigate if there is any difference
between the COVID-19 mortality rate in the US and the UK. On April 18, 2020, out of
a sample of 671,493 cases of COVID-19 in the US, there was 33,288 deaths; and out of
a sample of 109,754 cases of COVID-19 in the UK, there was 14,606 deaths. What is
the 92% confidence interval in the true difference in the mortality rates between the two
countries? What can you conclude about the difference in the mortality rates between
the US and the UK? Justify your conclusions. [8 marks]

The percentage of students with a GPA of 3.0 or higher is 15%. Suppose the random variable X represents the total number of students with a GPA of 3.0 or higher in a random sample of 500 students.

a) Find the mean of X. (round to the nearest whole number)

b) Find the standard deviation of X. (round to the nearest whole number)

c) Determine the shape of the distribution for X. (letter only)

A) skewed-left, since p > .5 and n is small.

B) skewed-right, since p < .5 and n is small.

C) bell-shaped, since p = .5 and n is small.

D) normal, since n ⋅ p ⋅ ( 1 − p ) ≥ 10.

d) Based on your answers from parts (a)-(c), would it be unusual for 80 students in the sample to have a GPA of 3.0 or higher. (letter only)

A) Yes, since P ( 80 ) < .05

B) Yes, since 80 is more than 2 standard deviations away from the mean

C) No, since 80 is within 2 standard deviations of the mean

D) No, since anyone can get a 3.0 or higher.

E) No, since P ( 80 ) ≥ .05.

e) Based on your answers from parts (a)-(c), would it be unusual for 95 students in the sample to have a GPA of 3.0 or higher. (letter only)

A) Yes, since P ( 95 ) < 0.05.

B) Yes, since 95 is more than 2 standard deviations away from the mean.

C) No, since 95 is within 2 standard deviations of the mean.

D) No, since anyone can get a 3.0 or higher.

E) No, since P ( 95 ) ≥ .05.

Although older Americans are most afraid of crime, it is young people who are more likely to be the actual victims of crime. It seems that older people are more cautious about the people with whom they associate. A national survey showed that 10% of all people ages 16-19 have been victims of crime.† At Jefferson High School, a random sample of

n = 68 students

(ages 16-19) showed that

r = 9

had been victims of a crime. Note: For degrees of freedom d.f. not in the Student's t table, use the closest d.f. that is smaller. In some situations, this choice of d.f. may increase the P-value a small amount and thereby produce a slightly more "conservative" answer.

(a) Do these data indicate that the population proportion of students in this school (ages 16-19) who have been victims of a crime is different (either way) from the national rate for this age group? Use

α = 0.05.

Do you think the conditions

np > 5

and

nq > 5

are satisfied in this setting? Why is this important?

(i) What is the level of significance?

What is the value of the sample test statistic? (Round your answer to two decimal places.)

(b) Find a 95% confidence interval for the proportion of students in this school (ages 16-19) who have been victims of a crime.

(c) How large a sample size should be used to be 95% sure that the sample proportion p̂ is within a margin of error

E = 0.05

of the population proportion of all students in this school (ages 16-19) who have been victims of a crime? Hint: Use sample data p̂ as a preliminary estimate for p. (Round your answer up to the nearest student.)
students

A researcher is interested in comparing the level of knowledge of U.S. History possessed by university students in various majors. Random samples of Health Science, Business, Social Science, and Fine Arts majors were selected and each student completed a high school senior level standardized U.S. history exam. The spreadsheet in the "US History Knowledge vs Student Major Dataset B" file on Blackboard for today's date contains data showing the test scores (percent correct) for each student by major. Assume that the test scores for students in each major are Normally distributed.

Confidence interval output from Minitab for this data is provided below

Individual 95% CIs For Mean Based on Pooled StDev Level N Mean StDev -------+---------+---------+---------+-- Health Science 8 56.63 18.93 (---------*--------) Business 8 48.75 20.10 (---------*--------) Social Science 8 70.25 16.27 (--------*--------) Fine Arts 8 56.50 21.61 (---------*--------) -------+---------+---------+---------+-- 45 60 75 90

Determine whether this data provides evidence that there are any statistically significant differences in the average test scores of students majoring in Health Science, Business, Social Science, and Fine Arts. Your answer should include:

- Mean test scores for students in each major.

- Discussion of the conditions, and why they are satisfied or why they are not satisfied. (Regardless of your answer here, assume the conditions are satisfied.)

- Hypotheses, test statistic, degrees of freedom, and P-value.

- Your conclusion regarding the question of whether there are any significant differences in the U.S. history knowledge levels of students with these four majors.

- Your assessment of the nature of any potential relationship(s) between the average U.S. history test score and student majors.

There are three ways a student can end up in Calculus II at Augusta University. They can either take Calculus I or place into the class via AP credit. Further, some students take Calculus I the semester immediately before Calculus II but some students wait one or more semesters between taking Calc I and Calc II. A professor has a Calculus II class with 21 students, and they look at students’ final grades and how they got into the class.

a. Carry out an ANOVA to see if there are any significant differences in the final grades among these three groups. Use ? = .05 to carry out the ANOVA, then use a suitably corrected alpha to do any followup tests if needed. Assume all the ANOVA assumptions (normal populations with equal variance) are satisfied.

b. A high school student coming to AU with AP credit is concerned about skipping Calculus I and wonders if they will be at a disadvantage relative to other students who take Calculus I right before taking Calculus II. Based on your analysis, is this concern warranted?

c. A student takes Calculus I in the fall and likes their professor, but this professor is not teaching Calculus II in the spring, so the student decides to wait and see if that professor is teaching the course at a later time. Based on the data, does this appear to be a sound course of action?

Q.1.Three students have each saved $1,000. Each has an investment opportunity in which he or she can invest up to $2,000. Here are the rates of return on the students’ investment projects:

Harry              5 percent

Ron                 8 percent

Hermione        20 percent

a. If borrowing and lending is prohibited, so each student uses only his or her saving to finance his or her own investment project, how much will each student have a year later when the project pays its return? [0.5 Marks]

b. Now suppose their school opens up a market for loanable funds in which students can borrow and lend among themselves at an interest rate r. What would determine whether a student would choose to be a borrower or lender in this market? [0.5 Marks]

c. Among these three students, what would be the quantity of loanable funds supplied and quantity demanded at an interest rate of 7 percent? At 10 percent? [0.5 Marks]

d. At what interest rate would the loanable funds market among these three students be in equilibrium? At this interest rate, which student(s) would borrow, and which student(s) would lend? [0.5 Marks]

e. At the equilibrium interest rate, how much does each student have a year later after the investment projects pay their return and loans have been repaid? Compare your answers to those you gave in part (a). Who benefits from the existence of the loanable funds market—the borrowers or the lenders? Is anyone worse off? [0.5 Marks] [Graph-0.5 Marks]

A researcher should use __________________ to analyze data when he has only a small sample, and he knows that the sample was drawn from a population that is seriously skewed.

Professor Smith has to grade assignments each week for her math class, and she suspects the week after her grading goes faster than usual. From previous experience, she knows the average amount of time it takes each week is 9 minutes per student and the standard deviation is 2.3 minutes per student. Assume a normal distribution. Last week, she had to grade assignments for 21 students. Let ?⎯⎯⎯⎯⎯ be the random variable representing the sample mean amount of time spent grading an assignment, for a sample of 21 students.

a. ?⎯⎯⎯⎯⎯ is normally distributed with a mean of _______________, and a standard error of the mean __________________. Round your answer to 2 decimal places.

b. Professor Smith finds that the week after midterms, her sample mean for 21 students is 7.44 minutes per student. Find the z-score associated to this sample mean, using the sampling distribution. Round your answer to two decimal places.

c. Find the probability that a randomly selected group of 21 students' assignments will take an average of 7.44 minutes or less to grade. Round your answer to 4 decimal places

d. Interpret the results. Is Professor Smith justified in concluding students' work is faster to grade the week after midterms?Select your answer from one of the following options.

• a.No, because probability is less than 0.05 that the average time per student was that low by chance
• b.No, because the probability is greater than 0.05 that the average time per student was that low by chance
• c.Yes, because probability is less than 0.05 that the average time per student was that low by chance
• d.Yes, because the probability is greater than 0.05 that the average time per student was that low by chance