Fall 2011 Math Placement Test for Haverford College

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Welcome to Haverford! This test will help us recommend math courses that seem right for you. We will make two recommendations: where to start in our "calculus sequence" and an appropriate statistics course if you decide to take statistics (either in addition to, or instead of, a calculus course). In late August, our recommendation will be given to your faculty advisor and (we hope) emailed directly to you.

Our recommendation is only a first step in finding an appropriate course for you. Your score on this test will not keep you out of any course you may want to take; it is just one tool to help you find a course that you are prepared for and will be challenged by. Once you arrive at Haverford, you will have several opportunities to discuss course choices with the math department, your advisor, etc.

The test is divided into five sections -- read the instructions for each section to see if you should fill out that section

Since the goal of the test is to gauge what Haverford math course would be best for you, don't guess if you have no idea of the correct answer to some question; just leave it blank. For the same reason, we see no need for you to study before taking the test. Finally,...

WHEN YOU ARE DONE, FIND ONE OF THE SUBMIT BUTTONS AND CLICK IT. (Yes, we already said this, but there's nothing worse than spending an hour-plus on the test and then failing to submit it...)

Section I: Background Questionnaire (For all students)

1. Do you think you have a solid background in basic high school math skills (algebra, geometry, trigonometry)?

Yes No Maybe

2. How much calculus have you had? None Less than a year A year or more

3. If you have had calculus, do you feel you know it reasonably well? Yes No Maybe

4. When was the last time you took math? What course?

5. If you took the AB Calculus Advanced Placement (AP) test, indicate the result:

Did not take Have not received results yet 5 4 3 2 1

6. If you took the BC Calculus Advanced Placement (AP) test, indicate the result:

7. If you took the Statistics Advanced Placement (AP) test, indicate the result:

8. Do you plan to take 3 or more semesters of math at Haverford? Yes No Maybe

9. Possible majors

Do you expect to major in math? Yes No Maybe

Do you expect to major in physics? Yes No Maybe

Do you expect to major in chemistry? Yes No Maybe

Do you expect to major in biology? Yes No Maybe

Do you expect to major in economics? Yes No Maybe

Do you expect to major in psychology? Yes No Maybe

Do you expect to pursue the pre-medical curriculum? Yes No Maybe

10. Have you taken a calculus or more advanced course at a college or university? Yes No

If yes, please indicate the class and institution:

11. Have you taken the International Baccalaureat Exam? Yes No

If yes, please indicate the results:

12. Please add any brief comments that you think might help us to recommending an appropriate placement in mathematics:

Section II: Pre-calculus questions (For all students)

1.

2.

3.

4.

4 100 5 20 None of these

5.

6.

1 2 3 4 5

7.

0 1 2 -2 -1

8. What is the slope of the line joining (0,3) to (6,0)?

1/2 -1/2 -2 2 18

9.

None of the above

10. An equation of the line passing through (-4,0) and (0,-2) is

11.

Section III: Derivative and integral questions (For students who have taken some calculus; if you have not taken any calculus, click Submit using the button above)

12.

13.

14.

15.

0 1 None of the these

16.

17.

18.

19.

The balloon reaches its maximum height at t=a The balloon is dropping fastest at t=b At t=c, the balloon is below its starting height (i.e., when t=0) At t=d, the balloon is above its starting height (i.e., when t=0) None of these statements is true

The balloon reaches its maximum height at t=a

The balloon is dropping fastest at t=b

At t=c, the balloon is below its starting height (i.e., when t=0)

At t=d, the balloon is above its starting height (i.e., when t=0)

None of these statements is true

20.

1/2 1/3 1/4 1 1/6

21.

22.

23.

0 1/8 1/4 -1/4 None of these

24.

25.

None of these

26.

Section IV below is just for students who have seen infinite series in some depth, e.g., in a Calculus BC course.

Section V below is just for students who have seen enough statistics that they might place past our introductory statistics course.

If you are in neither of these categories, then stop now and submit the test using the button below

Section IV: Infinite Series questions (For students who have seen this material, e.g., in a Calculus BC course; if you have not seen this material, please scroll down to Section V)

27.

converges to 3/2 converges to 2 converges to 0 converges to 5/3 does not converge

28.

converges to 3/2 converges to 2 converges to 0 converges to e does not converge

29.

converges to a value less than 0.1 converges to a value between 0.1 and 1 converges to a value between 1 and 2 converges to a value greater than 2 does not converge

converges to a value between 0.1 and 1

converges to a value between 1 and 2

converges to a value greater than 2

does not converge

30.

31.

1 sin(1) cos(1) e does not converge when x=1

32.

Section V: Statistics (Fill out this section only if you have taken enough statistics that you think you might place past our introductory statistics course, e.g., you took and did well in an AP-level Statistics course).

Within Section V, you may use a calculator. You should also use a "standard normal table", either your own or the one that can be downloaded by clicking this link: Standard Normal Table

33. A description of different houses on the market includes the following three variables. Which of the variables is quantitative?

The square footage of the house The monthly gas bill The monthly electric bill All of the above

Setup and Figure for Question 34 (five subquestions): In a statistics class with 136 students, the professor records how much money each student has in their possession during the first class of the semester. The histogram shown below represents the data collected:

34A. What (approximately) is the percentage of students with under $10 in their possession? 35% 40% 45% 50% 34B. Which of the following description(s) is/are correct regarding the shape of the histogram? Skewed right Skewed left Symmetric All of the above None of the above 34C. True or False? The histogram indicates the presence of an outlier. True False 34D. What is the number of students with $30 or more in their possession? Less than 5 About 10 About 30 More than 100 34E. From the histogram, which of the following is true? The mean is larger than the median The mean is smaller than the median The mean and median are equal It is impossible to compare the mean and median given the information available

34A. What (approximately) is the percentage of students with under $10 in their possession?

35% 40% 45% 50%

34B. Which of the following description(s) is/are correct regarding the shape of the histogram?

Skewed right Skewed left Symmetric All of the above None of the above

34C. True or False? The histogram indicates the presence of an outlier.

True False

34D. What is the number of students with $30 or more in their possession?

Less than 5 About 10 About 30 More than 100

34E. From the histogram, which of the following is true?

The mean is larger than the median The mean is smaller than the median The mean and median are equal It is impossible to compare the mean and median given the information available

The mean is larger than the median

The mean is smaller than the median

The mean and median are equal

It is impossible to compare the mean and median given the information available

35. Many residents of suburban neighborhoods own more than one car but consider one of their cars to be the main family vehicle. The age of these family vehicles can be modeled by a normal distribution with mean 2 years and standard deviation 6 months. What percentage of family vehicles is between 1 and 3 years old?

About 68% About 95% About 99.7% Cannot be determined based on the information given

Setup for Question 36 (four subquestions): Central Middle School has calculated a 95% confidence interval for the mean height of 11-year-old boys at their school and found it to be 56 2 inches.

36A. True or False: There is a 95% probability that is between 54 and 58. True False 36B. True or False: There is a 95% probability that the true mean is 56, and there is a 95% chance that the true margin of error is 2. True False 36C. True or False: If we took many additional random samples of the same size and from each computed a 95% confidence interval for , approximately 95% of these intervals would contain . True False 36D. True or False: If we took many additional random samples of the same size and from each computed a 95% confidence interval for , approximately 95% of the time would fall between 54 and 58. True False

36A. True or False: There is a 95% probability that is between 54 and 58.

36B. True or False: There is a 95% probability that the true mean is 56, and there is a 95% chance that the true margin of error is 2.

36C. True or False: If we took many additional random samples of the same size and from each computed a 95% confidence interval for , approximately 95% of these intervals would contain .

36D. True or False: If we took many additional random samples of the same size and from each computed a 95% confidence interval for , approximately 95% of the time would fall between 54 and 58.

37. To assess the accuracy of a laboratory scale, a standard weight that is known to weigh exactly 1 gram is repeatedly weighed a total of n times and the mean is computed. Suppose the scale readings are normally distributed with unknown mean and standard deviation = 0.01 g. How large should n be so that a 95% confidence interval for has a margin of error no larger than 0.0001 g?

n=100 n=196 n=10000 n=38416

38. The scores on the Wechsler Intelligence Scale for Children (WISC) are thought to be normally distributed with standard deviation = 10. A simple random sample of 25 children is taken, and each is given the WISC. The mean of the 25 scores is = 104.32. Based on these data, what is a 95% confidence interval for ?

104.32 0.78 104.32 3.29 104.32 3.92 104.32 19.60

39. In tests of significance about an unknown parameter, what does the test statistic represent?

The value of the unknown parameter under the null hypothesis The value of the unknown parameter under the alternative hypothesis A measure of compatibility between the null and alternative hypotheses A measure of compatibility between the null hypothesis and the data

The value of the unknown parameter under the null hypothesis

The value of the unknown parameter under the alternative hypothesis

A measure of compatibility between the null and alternative hypotheses

A measure of compatibility between the null hypothesis and the data

40. The square footage of the several thousand apartments in a new development is advertised to be 1250 square feet, on average. A tenant group thinks that the apartments are smaller than advertised. They hire an engineer to measure a sample of apartments to test their suspicions. Let represent the true average area (in square feet) of these apartments. What are the appropriate null and alternative hypotheses?

H_{0}: = 1250 vs. H_{a}: < 1250 H_{0}: = 1250 vs. H_{a}: 1250 H_{0}: = 1250 vs. H_{a}: > 1250

H_{0}: = 1250 vs. H_{a}: < 1250

H_{0}: = 1250 vs. H_{a}: 1250

H_{0}: = 1250 vs. H_{a}: > 1250

41. A college student is doing some research on the cost of one-bedroom apartments in town. He has randomly selected 25 apartments for which the price was published. The average price for these apartments is $652. He will assume that price follows roughly a normal distribution. Based on prices from previous years, a real estate agent gives him the information that is approximately $55. A 95% confidence interval for the true average price is found to be 652 21.56 = ($630.44,$673.56). Determine which of the following statements is true.

A test of the hypotheses H_{0}: = 650 vs. H_{a}: 650 would reject the null at the 0.05 level. A test of the hypotheses H_{0}: = 650 vs. H_{a}: > 650 would reject the null at the 0.05 level. A test of the hypotheses H_{0}: = 675 vs. H_{a}: 675 would reject the null at the 0.05 level. All of the above

A test of the hypotheses H_{0}: = 650 vs. H_{a}: 650 would reject the null at the 0.05 level.

A test of the hypotheses H_{0}: = 650 vs. H_{a}: > 650 would reject the null at the 0.05 level.

A test of the hypotheses H_{0}: = 675 vs. H_{a}: 675 would reject the null at the 0.05 level.

All of the above

42. A test of significance for a null hypothesis has been conducted and the p-value determined. Which of the following statements about a p-value is true?

The p-value is the probability that the null hypothesis is false. The p-value is the probability that the alternative hypothesis is true. The p-value is the probability that the null hypothesis is rejected even if that hypothesis is actually true. The p-value tells us the strength of the evidence against the null hypothesis. The larger the p-value, the stronger the evidence against the null hypothesis. More than one of these statements is true. None of these statements is true.

The p-value is the probability that the null hypothesis is false.

The p-value is the probability that the alternative hypothesis is true.

The p-value is the probability that the null hypothesis is rejected even if that hypothesis is actually true.

The p-value tells us the strength of the evidence against the null hypothesis. The larger the p-value, the stronger the evidence against the null hypothesis.

More than one of these statements is true.

None of these statements is true.

Setup for Question 43 (four subquestions): A small company consists of 25 employees. As a service to the employees, the company arranges for each of the employees to have a complete physical exam for free. Among other things, the weight of each employee is measured. The mean weight is found to be 165 pounds. The standard deviation of the weight measurements is 20 pounds. It is believed that a mean weight of 160 pounds would be expected for this group. To see if there is evidence that the mean weight of the population of all employees of the company is significantly larger than 160, the hypotheses H_{0}: = 160 vs. H_{a}: > 160 are tested. You obtain a p-value of about 0.106.

43A. True or False: At the 5% significance level, you have proved that H_{0} is true. True False 43B. True or False: You have failed to obtain any evidence for H_{a} True False 43C. True or False: At the 5% significance level, you have proved that H_{0} is true and a larger sample size will be needed to do so. True False 43D. True or False: The difference between the observed 165 pounds and the believed 160 pounds is very significant. True False

43A. True or False: At the 5% significance level, you have proved that H_{0} is true.

43B. True or False: You have failed to obtain any evidence for H_{a}

43C. True or False: At the 5% significance level, you have proved that H_{0} is true and a larger sample size will be needed to do so.

43D. True or False: The difference between the observed 165 pounds and the believed 160 pounds is very significant.

44. True or False: The power of a hypothesis test will increase when we increase the significance level .

45. True or False: The power of a hypothesis test will increase when we increase the sample size.

46. True or False: The power of a hypothesis test will increase when we consider an alternative value that is farther from the null value (the parameter of interest under the null hypothesis).

47. True or False: The power of a hypothesis test will increase when we increase the standard deviation.

Setup for Question 48 (three subquestions): A simple random sample of 120 vet clinics in the Midwest reveals that 32 of them treat large animals (cows, horses, etc.). Let p be the population proportion of vet clinics that treat large animals, and the sample proportion of vet clinics that treat large animals.

48A. Which is closest to the value of the standard error of ? 0.02 0.03 0.04 0.05 48B. What is a 90% confidence interval for p? (0.163,0.371) (0.188,0.346) (0.200,0.333) (0.667,0.800) 48C. If a 95% confidence interval were calculated instead, what would happen to the width of the confidence interval? It would be narrower It would stay the same It would be wider This can not be determined from the information given

48A. Which is closest to the value of the standard error of ?

0.02 0.03 0.04 0.05

48B. What is a 90% confidence interval for p?

(0.163,0.371) (0.188,0.346) (0.200,0.333) (0.667,0.800)

48C. If a 95% confidence interval were calculated instead, what would happen to the width of the confidence interval?

It would be narrower It would stay the same It would be wider This can not be determined from the information given

Thank you! If you have any technical difficulties, please email Rob Manning at rmanning@haverford.edu