Practice Questions for Test 1

Math 251, Spring 2004

Practice Questions for Test 1

Questions included here are some of the types of problems that could be on your first test. For further review, see the assignments from Chapters 1 through 4, as well as the quizzes and in-class exercise from those chapters and also the chapter review problems from Chapters 1 to 4 in the text. You may also wish to look at old tests (the topics will be slightly different, but you should see familiar types of questions on Tests 1 and part of Tests 2 from previous quarters. This test will be somewhat longer because of the two-hour class period.

1. Categorize the following data according to level: nominal, ordinal, interval, or ratio.

(a) Time of first class.

(b) Length of time to complete an exam.

(d) The undergraduate major of a prospective employee.

2. To estimate the average GPA of all La Sierra Students, President Geraty computed the average GPA obtained in his Advanced Hebrew Grammar class.

(a) Identify the variable?

(b) What is the implied population?

(d) What type of sample was this?

3. (a) If your instructor were to compute the class mean of this test when it is graded, and use it to estimate the average for all tests taken by this class this quarter, would this be an example of descriptive or inferential statistics? Explain.

(b) A study on attitudes about smoking is conducted at a college. The students are divided by class, and then a random sample is selected from each class. What type of sampling technique is this (e.g. simple random, convenient, stratified, systematic, cluster)? Explain why this type of sample is not a simple random sample.

(c) A politician wishes to determine determine the reading level of 5th graders in her State. She does not have funding to test all 5th graders in her state, so she randomly selects some of the schools in her state and tests every 5th grader in those schools. What type of a sample is this?

4. Classify the type of sampling used in the following examples.

(a) To conduct a poll, the Join Arnold team randomly chose 8 different prefixes in California (the first 3 digits of the telephone number) and called all households from those prefixes.

(b) To maintain quality control, a tire manufacturer tests every 100th tire that comes off of the assembly line in its plant.

(c) To determine student attitudes toward worship requirements at La Sierra, President Geraty gave questionnaires to ten randomly selected students each from of the following groups: Freshmen, Sophomores, Juniors, Seniors and Graduate Students.

(d) To determine worker attitudes in the recent MTA strike, the union numbered all employees, and used a random number generator to select 100 of those employees for interviewing.

(e) To estimate the mean age of students at La Sierra University, President Geraty computed the mean age of his Gender Issues in Ancient Hebrew Society class.

5. (a) Explain how you could use the table of random numbers in your text to help design a true false test of 10 questions so that the pattern of answers is random.

(b) How would you use a randomized two-treatment experiment in the following setting. Is placebo being used or not? Explain.

A veterinarian wants to test a strain of antibiotic on calves to determine their resistance to common infection. In a pasture are 22 newborn calves. There is enough vaccine for 10 calves. However, blood tests to determine resistance to infection can be done on all calves.

6. Consider the data:

100,102,104,108,108,110,110,112,112,112,115,116,116,118,118,

118,118,120,120,126,126,126,128,128,128,130,130,130,130,130,

132,132,134,134,136,136,138,140,140,146,148,152,152,152,156,

160,190,200,208,208

(a) What class width should be chosen if you would like to have 6 classes.

(b) Suppose you don’t want a class width of 19, but would like a class width of 15 irrespective of how many classes that would give you. Complete the following table for this data.

Lower

Limit

/ Upper

Limit

/ Lower Boundary / Upper Boundary /

Midpoint

Frequency

/ Cumulative

Frequency

/ Relative Frequency
99.5 / 114.5

(d) Draw a frequency polygon using the table in (b).

(e) Draw an Ogive using the table in (b).

7. For this question, consider the ogive that is displayed below for winning times for the Kentucky Derby.

(a) How many races had winning times under 2:05.15?

(b) What percentages of the races had winning times between 2:01.15 and 2:03.15?

8. The following represent the high temperatures in Celsius in Minneapolis over a 12 day period in January: -10, -8,-7,-7,-7,-4,-3,-3,-2,0,1,5

Find the mean, median and mode of these temperatures. Note  x = -45.

9. (a) Find the median of the sample: 5,6,8,9,11,1400,1401,2000,4837.

(b) In what place (or average of which places) would the median be given a sample size of:

(i) 1000 (ii) 2125

(c) Given a set of data of size at least 3, how does increasing the largest number affect the median? How would it affect the mean?

10. (a) Make a stem and leaf display for the following data.

5852688672669789849191

9266688786736170757273

85849057777684935847

(b) After making the display, find the median of the data.

11. Consider the two populations as follows:

Population 1: -5,-2,3,5,7,10,15,17,19,23,25,26,28,30,33,37,43,48,50, 56

Population 2: -9,4,8,13,18,22,25,30,38,39,42,51,58,70,78,79,97,106,108,119

(a) Which population do you think has a larger standard deviation? Why?

(b)Given that  x = 468 and  (x-)2= 5896.8 for Population 1, compute its standard deviation.

(d) Assuming Population 1 is merely a sample, compute the standard deviation:

(e) According to Chebyshev’s theorem, what proportion of the data in Population 2 should lie within 3 standard deviations of the mean. Compute the interval for Population 2 that is within 3 standard deviations of its mean.

12. Consider the following data of 26 numbers.

8 35 47 48 51 57 6064 64 65

66 70 72 7678 80 82 84 85 89

90 90 939496111

(a) Find the median of the data.

(b) Find Q1, Q3 and the IQR. Construct a box and whisker plot for the data.

(c) Compute the interval (Q1 - 1.5IQR, Q3 + 1.5IQR). Data outside of this interval are identified as suspected outliers. Are there any suspected outliers in the above data?

13. The following are ground water depths for a sample of 20 wells.

Distance from ground to water level (ft), x / 12—14 15—17 18—20 21—23 24—26
Number of wells, f / 1 3 8 2 6

Using the midpoints of the intervals, estimate the mean depth, the standard deviation, and the coefficient of variation.

14. (a) If you are among 8000 people that took a test and you scored at the 79th percentile, approximately how many people scored at your score or lower? Approximately how many people scored at your score or higher?

(b) If you are among 4000 students taking the MCAT, and you wish to score at least the 95th percentile, what is the maximum number of students that can score at least as well or better than you?

15. The following represents the outcomes of a flu vaccine study.

Got the Flu / Did not get Flu / Row Total
No Flu Shot / 223 / 777 / 1000
Given Flu Shot / 446 / 1554 / 2000
Column Total / 669 / 2331 / 3000

Let F represent the event the person caught the flu, let V represent the even the person was vaccinated, let H represent the event the person remained healthy (didn’t catch the flu), and let N represent the even that the person was not vaccinated.

(a) Compute: P(F), P(V), P(H), P(N), P(F given V) and P(V given F), P(V and F), P(V or F).

(b) Are the events V and F mutually exclusive? Are the events V and F independent? Explain your answers.

16. (Chapter Review Problem #8, p. 199) Class records at Rockwood College indicate that a student selected at random has a probability 0.77 of passing French 101. For the student who passes French 101, the probability is 0.90 that he or she will pass French 102. What is the probability that a student selected at random will pass both French 101 and French 102?

17. The following represents a group of 200 people attending a Dental Convention.

Has Beard / No Beard / Row Total
Male / 16 / 80 / 96
Female / 0 / 104 / 104
Column Total / 16 / 184 / 200

Let F be the event the attendee is female, M the event the attendee is male, B be the event the attendee has a beard, and N be the event the attendee has no beard.

(a) Compute: P(B), P(B given F), P(B given M).

(b) Are the events B and F independent? Are the events B and F mutually exclusive?

18. (Almost Question 24, p. 180 ) At Litchfield College of Nursing, 85% of incoming freshmen nursing students are female and 15% are male. Recent records indicate that 70% of the entering female students will graduate with a BSN degree, while 90% of the male students will obtain a BSN degree. In an incoming nursing student is selected at random, find

(a) P(student will graduate, given student is female)

(b) P(student will graduate, and student is female)

(d) P(student will graduate, and student is male)

(e) P(student will graduate)

(f) P(student will graduate, or student is female)

19. Suppose a 30km bicycle race has 28 entrants. In how many ways can the gold, silver and bronze medals be awarded.

20. (a)President Geratyhas recently received permission to excavate the site of an ancient palace. In how many ways can he choose 10 of the 48 graduate students in the School of Religion to join him?

(b) Of the 48 graduate students, 25 are female and 23 are male. In how many ways can President Geraty select a group of 10 that consists of 6 females and 4 males?

(c) What is the probability that President Geraty would randomly select a group of 10 consisting of 6 females and 4 males?

21. A local pizza shop advertises a different pizza for every day of your life. They offer 3 choices of crust style (pan, thin or crispy), 20 toppings of which each pizza must have 4, and 5 choices of cheese of which each pizza must have 1. Is their claim valid?

22. (a) How many different license plates can be made in the form xzz-zzz where x is a digit from 1 to 9, and z is a digit from 0 to 9 or a letter A through Z?

(b) What is the probability that a randomly selected license plate will end with the number 00? That is the license plate looks like xzz-z00?

Some True/False Questions

23. Determine whether the following statements are True or False.

(a) Mutually exclusive events must be independent because they can never occur at the same time.

(b) Independent events must be mutually exclusive because they are independent from one another.

(d) If A and B are independent, then P(A and B) = P(A)  P(B).

(e) The probability that an event A does not occur is 1 – P(A).

(f) If A and B are independent, then P(A or B) = P(A) + P(B).

(g) When we consider permutations, we consider groupings but not order.

(h) When A and B are mutually exclusive events, the formula:

P(A or B) = P(A) + P(B) – P(A and B)

simplifies to P(A or B) = P(A) + P(B) because P(A and B) = 0.

(i) Suppose A is the event that an ace is the first card drawn from a deck of cards, and let B be the event that a King is the second card drawn out of a deck of cards without replacement. Then A and B are independent because different cards are to be drawn.

24. Determine whether the following statements are True or False.

(a) In a frequency polygon, a dot whose height represents the class frequency is placed above the upper class boundary.

(b) In an ogive, a dot whose height represents the cumulative frequency of the class is placed above the class mark.

(d) The class limits of a frequency distribution are allowed to be data from the distribution.

(e) The class boundaries of a frequency distribution are not allowed to be data values from the distribution.

(f) A pareto chart is a bar graph whose bars are arranged from left to right according to increasing height.

(g) If a frequency distribution with 9 classes is to be constructed from a distribution that includes data values from 0 to 90 including 0 and 90, then the class width should be 90/9 = 10.

(h) The distinguishing difference between a relative frequency histogram and a frequency histogram is that in a relative frequency histogram, the bars’ heights are the relative frequencies of the class rather than the frequencies of the class.

(i) In a skewed distribution, the direction of skewness is on the side with the longer tale.

(j) A symmetric distribution is not skewed.

(k) In an ogive, the height of the dot at the upper boundary minus the height of the dot at the lower boundary is equal to the class width.

25. Determine whether the following statements are true or false.

(a) Chebyshev’s theorem says that in any distribution, exactly 75% of the data is within 2 standard deviations of the mean.

(b) The 2nd quartile is the median.

(d) In a box plot, 75% of the data is at or above the bottom line on the box.

(e) If the sample standard deviation formula and population standard deviation formula were computed using exactly the same data, the population standard deviation formula would give a larger number.

(f) The coefficient of variation is defined as the mean over the variance.