Exam Prep: Selected Test Questions (Statistics) Unit 1: One-Variable Statistics 1. What type of variable is “shoe size”? a) Qualitative/categorical, nominal b) Qualitative/categorical, ordinal c) Quantitative/numerical, discrete d) Quantitative/numerical, continuous 2. Socioeconomic status, recorded as “low income”, “medium income” or “high income” is considered what type of data? a) Qualitative/categorical, nominal b) Qualitative/categorical, ordinal c) Quantitative/numerical, discrete d) Quantitative/numerical, continuous 3. Students’ favourite colours are shown. Which measure of center best represents the data? a) Mean b) Median c) Mode d) Either a or b 4. The histogram below shows the number of minutes needed by students to complete a computer game. What is the shape of this distribution? a) Uniform b) Symmetrical c) Left-skewed d) Right-skewed 5. What is not something stem plots and histograms have in common? a) they show the shape of a distribution b) they display quantitative data only c) they group data together d) they display individual data values 6. A z-score tells you a) how far above or below the mean a score lies b) if there are outliers c) how many standard deviations a data value is above or below the mean

Colour Blue Pink Purple Red Orange

Frequency 12 6 4 2 2

d) if the distribution is symmetrical or skewed 7. Data were collected on the amount, in dollars, that individual customers spent on dinner in an Italian restaurant. The quartiles for these data are given below. Which of the following statements must be true for these Q1 Q2 Q3 customers? $36.27 $44.27 $58.97 a) At least half of the customers spent less than or equal to $44.27 and at least half spent greater than or equal to $44.27. b) 75% of the customers spent between $36.27 and $58.97. c) 25% of the customers spent less than or equal to $58.97 and 75% spent greater than or equal to $58.97. d) The majority of customers spent $44.27. 8. The box plot below shows the age distribution of a random sample of 374 pennies. The approximate interquartile range (IQR) of the pennies is a) 10 b) 16 c) 40 d) 50

Short Answer 9. Compare bar graphs to histograms. What is similar? What is different? Consider the data they display in your answer. [K/C 2] 10. Explain the problems with each graph. [K/C 3] a)

b)

2000

2015

11. The mean mark of a MDM4U class of 20 students is 65%. The mean mark of another MDM4U class of 30 students is 75%. What is the combined mean of the classes? [A 1] 12. The salaries of employees of a company are shown below. Determine

[K 4]

Salary

$16 500

$22 300

$30 100

$34, 800

$42 500

$51 100

$130 300

Number of Employees

5

10

6

4

2

2

1

a) the mode

b) the median

c) the mean

13. Baldo’s pizza restaurant is monitoring its delivery process. The restaurant randomly selects a sample of five pizzas one night and records their delivery times, in minutes: 28

32

50

40

35

a) Find the mean delivery time. [K 1] b) Find the standard deviation. [K 3] c) Pizzazz Pizza across the street has a mean delivery time of 36.2 minutes and a standard deviation of 5.5 minutes. Compare the delivery service of these two restaurants. [A 1] d) Is it appropriate to compare these two restaurants in this way? Explain. [T 1] 14. The following are IQ scores of 30 sixth graders. 90 95 a) b) c) d) e) f)

97 98

151 122 102 88 81 102 105 109

93 84

109 117 114 107 99 119

87 81

99 94

115 126 92 102 94 101

Create a stem plot. [K 3] Determine the five-number summary. [K 2] Determine if there are outliers. [K 2] Create a box plot of the IQ scores. [K 3] What measures of center and spread are most appropriate for this data set? Why? Describe the distribution of IQ scores, using SOCS as a guideline and appropriate numerical measures (for center, spread). [C/A 2]

15. In university, a grade point average (GPA) of your grade is computed, which is a weighted mean of your course marks. Marks range from 0.00 (F) to 4.33 (A+). Courses are weighted differently depending on how many credits they are worth. Calculate the GPA of a student who earned the following grades in each of five courses.

Course

Credits

PYSCH 100 ENGL 015 CHEM 012 MATH 040 KINES 048

3 3 3 5 1

Letter Grade C B+ F BA

Mark 2.00 3.33 0.00 2.67 4.00

Unit 2: Collecting Data & Two-Variable Statistics 1. Data that is used by someone who did not collect the data is a) primary data b) secondary data c) microdata d) aggregate data 2. Which type of data includes the individual responses to surveys rather than a summary? a) primary data b) secondary data c) microdata d) aggregate data 3. A correlation coefficient of –0.8 indicates what strength of correlation between two variables? a) no correlation b) weak correlation c) moderate correlation d) strong correlation 4. Double-blinding is a technique using in experiments to a) eliminate sampling bias b) eliminate observation bias and experimenter bias c) ensure that subjects are randomly assigned to a treatment or control group d) prevent the Hawthorne effect 5. People with arthritis are asked if they have had acupuncture therapy and whether it reduced their pain. This is an example of a) an observational study b) an experimental study c) the Hawthorne effect d) a random sample 6. A survey for Woodland students asks if they go to church regularly. What type of bias is possible? a) response bias b) non-response bias c) voluntary response bias d) sampling bias

7. The most likely r value of the following graph is a) 0 b) 0.2 c) 0.7 d) 1 8. The best way to determine cause and effect is to: a) perform a linear regression analysis b) find a high correlation coefficient c) do an observational study d) do an experiment 9. Identify the type of sampling used (simple random, systematic, stratified, cluster, voluntary, convenience). [3] a) A market research company mails surveys to all of the adult residents in a town and asks residents to mail back their responses in a prepaid envelope. ______________________________ b) Researchers want to investigate the use of pesticides by apple farmers in Ontario. They divide the province into 10 sections and sample all farms within 5 randomly chosen sections. ______________________________ c) Students who have been assigned numbers are chosen to participate in a survey using a random number generator. ______________________________ 10. Compare stratified sampling and cluster sampling. What is similar? What is different? Include at least three points. [3] 11. The marketing department for a large retail chain wants to survey its customers about a new advertising plan. They go into one of their largest Toronto stores on a Tuesday morning and survey the first 50 people who make a purchase. [3] a) Identify the sampling method used: ____________________________ b) Is this method likely to provide a representative sample? Why or why not? c) How can the sampling method be improved? 12. A medical researcher wants to test a new drug in nicotine patches believed to help smokers overcome their addiction. Twenty people who want to quit smoking volunteer for the study. The smokers are given the choice to take the regular nicotine patches or the patches with the new drug. Two of the eight who used the regular patches quit smoking and seven out of twelve who used the ones with the new drug quit smoking. Can the researcher conclude that the new drug is effective? In your answer, include a critique of the design of the experiment. [3]

13. Read the comic from Family Circus where Billy is flying on a plane. [2] a) What type of relationship does Billy understand there to be between the seatbelt sign and bumpiness? Explain. b) What relationship is actually present? Explain.

14. Liam wants to explore student’s health habits. He designs an online survey and sends the survey to every tenth student on the alphabetical student list through email. His questions include the following: 1. How many hours, on average, do you exercise in a week (in a gym, playing sports)? a) 0 hrs b) 0 – 2 hrs c) 2 – 4 hrs d) 4 – 6 hrs e) 6 – 8 hrs f) 8 – 10 hrs g) more than 10 hrs 2. How regularly do you get proper nutrition and sleep? a) Very rarely b) Rarely c) Sometimes

d) Often

e) Very often

3. It is estimated that 1 in 10 children are clinically obese. Do you think physical exercise should be increased in schools? a) Strongly disagree b) Disagree c) Agree e) Strongly agree a) What type of sampling method did Liam use? _____________________ [1] b) What bias might occur due to the way Liam collected his sample? Note: do not refer to survey questions here. [1] c) What are the problems with Liam’s survey questions? Suggest how to fix them. [4] 15. A student is conducting a survey about involvement in sports activities. The school of 800 students has the following breakdown. a) Determine how to obtain a sample of 100 students stratified by grade. b) Determine how to obtain a sample of 100 students stratified by gender.

Grade 9 10 11 12

Gender Female Male Female Male Female Male Female Male

Number of Students 120 180 125 125 100 50 30 70

16. Consider the graph below. Can we conclude that firefighters cause more damage to burning homes? Explain. [2] 17. A university professor tracks the mean final grade and class size of her first-year statistics classes over a number of years to determine if class size affects student performance. The results of the professor’s analysis are shown below.

r = –0.81 G = –0.092s + 79.285 (G: mean grade (%), s: class size

a) b) c) d) e) f) g)

Identify the independent and dependent variables. Describe the correlation in words (“As…”). Interpret the correlation coefficient (r value). Interpret the regression equation (line of best fit). Predict the mean final grade for a class size of 40 students. Are you interpolating or extrapolating in question e? What can the professor conclude? Explain.

ANSWERS Unit 1: One-Variable Statistics 1. c 2. b 3. c (it’s categorical data) 4. c 5. d (histograms do not display individual data) 6. c 7. a 8. B 9. Similarities: Histograms and bar graph can both display quantitative discrete data and have bars. Differences: only bar graphs can display qualitative data, while histograms can handle a lot of data and data that is continuous; with histograms, data is grouped into intervals so individual data values are not shown. 10. a) Minor problems: missing title and label on vertical axis; Major: The 3D effect makes it difficult to read the graph (keep it simple) and the axis is truncated, meaning it doesn’t start at 0, so differences are exaggerated. b) This pictograph makes the house prices look much more than twice as much in 2015 than 2000 because of the scaling of the house. A simple bar graph would show the numbers more fairly. 11. 71% (use a weighted mean) 12. a) mode: $22 300 b) median: $26 200 (between the 15th and 16th data values) c) mean: $31 426.67 13. a) 37 min b) 8.49 min c) They are fairly similar in terms of typical delivery time (means are close) but Pizza Pizzazz is more consistent with their delivery times (less spread out) due to a lower standard deviation. e) No; these numbers are based on only 5 delivery times. Plus, there’s no indication of how far each location was from the pizza shop. 14. a) See stem plot below left b) (81, 93, 100, 109, 151) c) lower fence: 69; upper fence: 133; yes, 151 is an outlier d) See box plot below right (it can be horizontal and you should have titles and a label on the axis; note the outlier is a separate point)

IQ Scores of Gr.6 Students

8 | 1 = score of 81 e) median and IQR due to the outlier (the outlier will greatly affect the mean, standard deviation and range). f) The IQ scores appear right-skewed (a histogram would confirm this), with an outlying score of 151. The typical IQ in the Gr.6 class is 100, which is the median score; half the scores are below this and half are above. The middle 50% of IQ scores have a range of 16 (the interquartile range). 15. 2.22

Unit 2: Collecting Data & Two-Variable Analysis 1. b 2. C 3. D 4. B 5. a 6. a 7. c 8. c 9. a) voluntary b) cluster (all are taken from some) c) simple random 10. Similarities: there are groups; they are random processes; differences: with cluster, groups are usually naturally formed by location, only some clusters/groups are chosen and all members in a cluster are part of the sample (All from SOME); with stratified, groups are created by a characteristic like gender or race, and members are taken from EACH group in proportion to the amount in the population (SOME from ALL). 11. a) convenience b) No – there are many reasons: only one store was part of the sample, with shoppers from Tuesday morning (which misses those who are working then) and they only sample people who make a purchase c) take a sample from various stores, on different days of the week, during various hours. Also sample from customers who don’t make purchases. 12. Consider: control, randomization, replication; The researcher did ensure control because s(he) had a group that took a regular patch (the control group) and a group that took a patch with the new medication (treatment group). The researcher did not randomly assign participants to the control or treatment group – they chose, so this could leave to observation bias (Hawthorne effect) and experimenter bias. (Instead, double-blinding should be done). Finally, 20 participants is not a large number; more replication should be done by including more participants or repeating the experiment. 13. a) Billy thinks that the seat belt sign causes bumpiness (cause and effect). b) The reverse is true; the bumpiness causes the seat belt sign to go on (reverse cause and effect; you could also say this is a common cause: turbulence 14. a) systematic sample b) non-response bias – people don’t respond c) #1: there are overlapping answers (how do you answer if you exercise 2 hours per week?) Separate the answers (i.e. 0 – 1, 2 – 3, 4 – 5, etc.) #2: There are 2 questions asked (double-barrelled question); ask the questions separately; “proper” nutrition and sleep can also be interpreted differently; define those; give a “don’t know” option? #3: This is a leading question due to the first sentence; take it out. Also, give a neutral option. 15. a) Gr.9: 330/800*100 = 38 students, Gr.10: 31 students, Gr.11: 19 students; Gr.12: 13 students (Note: this makes 101, but that’s ok. It’s just due to rounding. b) female: 41 students; male: 59 students 16. No; there is a common cause; larger fires will cause both more fire fighters to be involved and larger damage. 17. a) independent: class size; dependent: mean final grade b) As class size increases, the class mean final grade decreases. c) There is a negative strong correlation between class size and mean final grade. d) As class size increases by one student, the grade decreases by 0.092 and with a class size of 0, the mean final grade is 79.3 (which is not very useful). e) 75.6% f) Extrapolating – 40 is below the lowest number of students in the class (outside data range). g) The professor should be cautious in concluding anything. Even though the r-value is high, this is a very small sample size. More students should be included in the study. Secondly, there are many hidden variables to consider, like student’s effort (studying, practice) and ability, attendance, extra help they may get (tutoring), etc. They should account for these variables in a new study with many more participants.