BU 255 Statistics · Midterm 1 Practice Questions

Start here: the exam and how to use this guide

Midterm 1 covers Module 1 only, on Friday October 23 from 7:30 to 9:30 PM, so every question below comes from descriptive statistics, probability, discrete distributions or continuous distributions. Nobody outside the exam can know the actual questions, so each one carries a likelihood tag based on how heavily the course materials drive it.

What the materials tell you about the exam

Item	What the course says
Coverage	Module 1: Black chapters 1-6 (parts 1-4 of the Module 1 slides)
Format	The posted materials do not describe the question format. The Module 1 Quiz is the mock exam and is multiple choice with 2 attempts
Concept checks	One instructor says in-class concept-check questions may reappear on exams, so the lesson concept checks are included here
Mock quiz	Your course grade uses the higher of the quiz grade and the exam grade, so take the Module 1 Quiz. MyLS closes it at 7:30 PM on Oct 23; the syllabus says the day before, so finish it early
Practice	WileyPLUS practice 1-1 to 1-4 (descriptive, probability, discrete, continuous) are ungraded and recommended
Normal table	The course z-table (Black Table A.5) gives the area between the mean and z
Support	The Second Year Support Centre runs Mon-Fri 4-7 PM, extended to 10 PM Tue-Thu in midterm weeks

Because the format is not stated, the questions here are written in the multiple-choice style with a calculation behind each answer. Practise them with the answer covered and the working on paper.

What is skipped

The syllabus skips the empirical rule, Chebyshev, counting rules, probability matrices, binomial and Poisson tables, and the Poisson approximation to the binomial. Among quantitative graphs only histograms are in scope. Percentiles, quartiles, the IQR and box plots appear in one instructor's slides while another lesson page says to skip them, so they are tagged Possible. Weighted means and grouped-data formulas do not appear in the posted notes.

How to read the likelihood tags

Tag	Meaning
Very likely	The standard example of the topic, or a concept check with an answer key, or practice repeated by more than one instructor
Likely	Worked in the slides with full numbers, or a natural variation of a very likely question
Possible	Appears in one instructor's slides only, or is on the edge of the syllabus

Each question has a stem, then How to answer (the method, in order), then Answer (the result and the one-line reason).

Pick the tool from the wording

If the question says	Use
Categories only, no order	Nominal; mode only; bar or pie chart
Rating or rank, unequal gaps	Ordinal; median and mode, never the mean
Counts in n fixed trials, two outcomes, constant p	Binomial: n, p, x
Count of events in an interval, constant rate	Poisson: rescale lambda to the interval, never change x
Time until or between events	Exponential: lambda is a rate, mean = 1/lambda
Every value in a range equally likely	Uniform: width ÷ total width
Bell-shaped measurement with a mean and SD	Normal: z = (x − mu) ÷ sigma, then the table
New evidence revises an earlier probability	Bayes: prior, conditional, joint, then joint ÷ total
Of those who..., among..., given...	Conditional probability: the group named is the denominator

Habits that save marks

Write the givens with symbols before calculating (n, p, lambda, mu, sigma, P(A), P(B|A)).
Draw the picture: a Venn diagram, a tree, or a bell curve with the shaded area.
Population uses N and sigma; a sample uses n − 1 and s. Read the wording.
Round z to 2 decimals for the table and keep 4 decimals for probabilities.
For discrete counts, fewer than 4 means at most 3; for continuous variables, < and ≤ give the same answer.
Use complements for at least and more than: P(X > 1) = 1 − P(0) − P(1).
Sanity-check every probability: it must lie between 0 and 1.

Part 1A: Data, measurement and graphs

This part tests vocabulary and judgment more than arithmetic, so the marks come from naming the level of measurement, the type of analytics and the right chart without hesitation. Sixteen questions follow.

Q1Very likely

Name the level of data for each question: (a) How many automobiles does your family own? (b) What is the temperature today in Celsius? (c) What is your annual income in dollars? (d) Are you married, yes or no? (e) Rate this product: Excellent, Good, Fair, Poor.

Q2Very likely

A table lists 8 employees with Case, Name, Age, Income, Position and Seniority Level (1 to 5). How many variables, elements and observations are there, and what is the level of each variable?

Q3Very likely

Six students rate exam difficulty from 1 (too easy) to 5 (too difficult): 2, 4, 1, 5, 3, 3. Which measures of central tendency are proper?

Q4Likely

Classify each statement as descriptive or inferential, and each number as a parameter or a statistic. (a) The mean age of all 289 people in the company database is 41. (b) A sample of 500 voters is used to predict the election result. (c) The sample mean income is $62,000.

Q5Likely

Classify each as descriptive, predictive or prescriptive analytics. (a) Last quarter's sales report by region. (b) A regression forecast of next quarter's sales. (c) A model that recommends the production mix that maximizes profit given machine limits.

Q6Likely

Match each V of big data to its meaning: Volume, Variety, Velocity, Veracity, Value.

Q7Very likely

Of 780 BU 255 students, 117 earned an A+ (the top class on the 0 to 12 Laurier scale). What are the relative frequency and cumulative frequency of A+?

Q8Very likely

Sixty years of Canadian unemployment fall into classes with frequencies 4, 12, 13, 19, 7 and 5 for 1-under 3, 3-under 5, 5-under 7, 7-under 9, 9-under 11 and 11-under 13. Build the relative and cumulative frequencies. What proportion of years had unemployment under 7%, and which class is modal?

Q9Likely

The 60 unemployment rates run from 3.6 to 12.0. You want 6 classes. What class width do you use, and how many classes does the guideline suggest for 40 observations?

Q10Likely

Using the same raw data, Black's table has classes 2-under 4 through 12-under 14 with frequencies 2, 10, 29, 11, 7 and 1. How many years were under 8%, what is the modal class, and what is the midpoint of the modal class?

Q11Very likely

Choose the best graph for each: (a) share of orders by food category; (b) delivery distance against delivery time; (c) monthly orders over 24 months; (d) the distribution of 60 yearly unemployment rates; (e) ranking defect types by how often they occur.

Q12Likely

Student spending totals $599.38, of which electronics is $211.89 and clothing and accessories is $134.40. Find each pie slice's percentage and angle.

Q13Likely

What is the difference between a histogram and a bar chart? What is the naming trap between instructors?

Q14Possible

Electric-motor problems: poor wiring 40, short in coil 30, defective plug 25, bearing seized 5. Draw the Pareto logic: which causes account for 70% of problems?

Q15Likely

House prices ($1000s) against size (sq ft) for ten homes show larger homes with higher prices. Monthly orders over two years show a dip in May to August each year and slightly higher values in year 2. Name the graphs and the patterns.

Q16Possible

Why can the same data look different in two histograms?

Part 1B: Central tendency, variability and shape

The calculation questions here reward one habit above all: decide whether the data are a population or a sample before dividing. Sixteen questions follow.

Q1Very likely

A plant made 5, 9, 16, 17 and 18 machines in five weeks (all the weeks of interest). Find the mean, the range, the variance and the standard deviation.

Q2Very likely

The same five numbers are a sample drawn from a longer production history. Find the sample variance and standard deviation. Why n − 1?

Q3Very likely

Revenues ($ millions) of a sample of six large accounting firms: 2,300; 1,446; 1,428; 1,397; 660; 610. Find the mean, sample variance and sample SD.

Q4Very likely

A boutique records 15 transactions: 24, 31, 45, 28, 35, 38, 19, 63, 41, 35, 29, 650, 47, 33, 26. Find the mean, median and mode. Which describes a typical sale, and what is the skew?

Q5Very likely

Thirty students answer, regular exercise is important for my mental health: strongly disagree 2, disagree 2, neither 8, agree 13, strongly agree 5. Which measure of centre should you use, and what is it?

Q6Very likely

Ten valves have a population mean weight of 100 g and SD of 20 g. One valve recorded as 100 g actually weighs 90 g. What happens to the mean and the SD?

Q7Very likely

Same valves, but one unidentified valve loses 10 g. What can you say about the mean and SD?

Q8Very likely

Stock X averages $12.20 with SD $1.00; stock Y averages $341.30 with SD $6.50. Which is riskier relative to its price?

Q9Likely

Two suppliers each average 6 days of delivery time over the last 10 deliveries. A: 6, 5, 7, 6, 5, 6, 7, 6, 5, 7. B: 3, 9, 4, 8, 2, 10, 5, 9, 4, 6. Which is more dependable?

Q10Likely

A distribution has mean 52, median 48 and mode 45. A second has mean 40, median 44, mode 47. Describe the shape of each.

Q11Likely

Every salary in a firm rises by exactly $2,000. What happens to the mean, the standard deviation and the CV? What if every salary is multiplied by 1.05?

Q12Likely

For a population of 5, the sum of x is 65 and the sum of x squared is 975. Find the variance without deviations.

Q13Likely

Marks: Anna scores 82 in a course with mean 75 and SD 5. Ben scores 70 in a course with mean 60 and SD 10. Who did better relative to their class? What score has z = −1.5 when the mean is 50 and the SD is 10?

Q14Likely

Which measure of centre fits each? (a) The most purchased category last month: apparel 6, appliances 3, books 4, computers 4. (b) Children per household recorded as 0, 1, 2, 3 or 4 or more. (c) Exam marks on a ratio scale with no extreme outliers.

Q15Possible

Order the data 106, 109, 114, 116, 121, 122, 125, 129. Find Q1, the median, Q3 and the IQR. Also, which positions do you average for the 80th percentile of 1,240 values?

Q16Possible

With Q1 = 111.5 and Q3 = 123.5, where are the inner and outer fences, and how would you classify observations of 150 and 170?

Part 2A: Probability rules, conditional probability and independence

This is the heart of Lesson 2. Almost every question reduces to picking the right one of five formulas, so the first table is worth memorising before you start.

The question says	Symbol	Formula
"not A"	P(A′)	1 − P(A)
"A or B (or both)"	P(A ∪ B)	P(A) + P(B) − P(A ∩ B)
"A and B"	P(A ∩ B)	P(A) × P(B \| A) in general, or P(A) × P(B) only if independent
"A given B", "of those who B"	P(A \| B)	P(A ∩ B) ÷ P(B)
independent test		P(A \| B) = P(A), or P(A ∩ B) = P(A)P(B)

Q1Very likely

Classify each as classical, relative frequency or subjective probability: (a) a 1,000-ticket raffle gives you one ticket, so your chance of winning is 1/1,000; (b) last election data show 30% of voters were younger than 35; (c) a doctor estimates a 70% chance that a patient returns to sport.

Q2Very likely

Ten customer orders: loyalty members (L) placed O1 to O4, and regular members placed O5 to O10. Orders O1, O2, O3, O5 and O6 were express shipping (E); the other five were ground. Find P(L), P(R) where R means regular, P(L ∪ E) and P(L ∩ E).

Q3Very likely

Using the same ten orders, find P(E | L) and P(L | E). Are they equal?

Q4Very likely

Four of the ten orders (O1, O2, O4, O7) are pulled for quality control (QC); three of those four are loyalty orders and two are express. Is QC independent of L? Is QC independent of E?

Q5Very likely

A used-car lot: 70% of cars have air conditioning (AC), 40% have Bluetooth (BT), and 20% have both. Find (a) P(no BT), (b) P(AC and BT), (c) P(AC or BT), (d) P(BT | AC), (e) P(AC | BT). (f) Are AC and BT independent?

Q6Very likely

43% of Canadians expect to save more next year (M) and 45% plan to reduce debt (R). Of those who expect to save more, 81% plan to reduce debt. (a) Which probability is the 81%? (b) Find P(M ∩ R). (c) Find P(M | R).

Q7Very likely

Roll one die. For each pair, decide if the events are mutually exclusive and/or independent: (i) {2 or more} and {even}; (ii) {4 or less} and {even}; (iii) {a 4} and {5 or less}; (iv) {a 6} and {5 or less}.

Q8Very likely

True or false: two events that are mutually exclusive (and each has a positive probability) must be independent.

Q9Very likely

A number from 1 to 10 is picked at random. X = even, Y = greater than 6. Find P(X ∪ Y).

Q10Very likely

Anna plays two separate card games. She wins the first with probability 0.10 and the second with probability 0.20, independently. What is the chance she wins at least one?

Q11Very likely

A standard 52-card deck. Find P(King), P(Red), P(King | Red) and P(Red | King), and decide whether King and Red are independent. Then find P(King of hearts | Red) and decide whether King of hearts and Red are independent.

Q12Very likely

From the same deck, find P(King or Red), P(King of hearts or Red), P(King or Queen), P(King and Red) and P(King of hearts and Red).

Q13Very likely

Fourteen people were asked about candy and soda. Two like both, four like candy only, five like soda only, and three like neither. A student says P(candy or soda) = 6/14 + 7/14 = 13/14. Is that right? Find the correct value and P(candy | soda).

Q14Very likely

A company has 140 employees. 80 are married and 30 are supervisors. 20% of married employees are supervisors. What is the probability that a randomly chosen employee is both married and a supervisor?

Q15Very likely

(a) Of 200 stadium staff, 90 are trained in crowd control (C), and 40% of those are also first-aid certified (F). Find P(C ∩ F). (b) 70% of orders are placed online (O), and 15% of online orders need an address correction (A). Find P(O ∩ A).

Q16Likely

32% of Canadian grocery chains have an ATM and 19% have a pharmacy; assume the two are independent. Find P(both). Separately, each item tested on a line is defective with probability 0.04, independently. Find P(the first two items are both defective) and P(at least one of the two is defective).

Q17Likely

In an office-design survey, 70% of staff chose noise reduction (N), 67% chose more storage (S), and 56% chose both. Find P(N ∪ S), P(S | N), and say whether N and S are independent.

Q18Likely

The banker data below classify 177 bankers by job satisfaction (1 to 5) and age. Find (a) P(satisfaction is 4 or 5), (b) P(under 30), (c) P(satisfaction 5 | over 50), (d) P(over 50 | satisfaction 5).

Satisfaction	Under 30	30 to 50	Over 50	Total
1	7	3	0	10
2	19	14	3	36
3	28	17	12	57
4	11	22	16	49
5	2	9	14	25
Total	67	65	45	177

Q19Possible

One die: X = {3, 6} and Y = odd = {1, 3, 5}. Find P(X), P(X′), P(X ∪ Y), P(X ∩ Y) and P(X | Y), and say whether X and Y are independent.

Q20Possible

A pair of dice is rolled. Event X: the first die shows 2. Event Y: the total of the two dice is 5. Event Z: the second die shows 5. Which pair is independent and which is dependent?

Part 2B: Bayes' rule and total probability

Bayes questions look hard but follow one fixed recipe. Use the table method every time, because it is the same four columns whatever the story is.

Event	[1] Prior P(E)	[2] Conditional P(A \| E)	[3] Joint = [1] × [2]	[4] Revised = [3] ÷ total of [3]
one row per cause	must sum to 1	given in the story	multiply across	divide each joint by the column total

The total of column [3] is the marginal probability P(A) (the law of total probability). Column [4] must sum to 1, which is your built-in check.

Q1Very likely

5% of transactions are fraudulent. The monitoring system flags 80% of fraudulent transactions and 10% of legitimate ones. A transaction is flagged. What is the probability it is fraudulent? Also find the probability a transaction is fraudulent given that it was not flagged.

Q2Very likely

A prospective MBA student figures he has a 10% chance of a high GMAT score (650 or more). Among MBA graduates with a high score, 52% took a prep course; among those with a lower score only 23% did. He takes the course. What is the probability he scores high?

Q3Very likely

A plant has two production lines. Line A makes 60% of output with a 2% defect rate; line B makes 40% with a 5% defect rate. A product is picked at random and is defective. What is the probability it came from line B? What is the overall defect rate?

Q4Very likely

Two suppliers provide an over-the-counter drug. Prairie supplies 65% of it and Badlands 35%. Side effects occur in 8% of Prairie's drugs and 12% of Badlands'. A customer reports a side effect. Find P(Prairie) and P(Badlands) after that news. Then find P(Prairie | no side effect).

Q5Very likely

A disease affects 1% of the population. A test detects the disease in 99% of people who have it, but also gives a false positive for 5% of healthy people. You test positive. What is the probability you have the disease, and why is it so low?

Q6Very likely

Using the ten customer orders (loyalty L, regular R; express E): P(E | L) = 0.75, P(E | R) = 2/6, P(L) = 0.4, P(R) = 0.6. (a) Find P(E) using the law of total probability. (b) Find P(L | E) using Bayes' rule.

Q7Likely

A factory has three shifts. Day makes 50% of output, evening 30% and night 20%, with defect rates of 1%, 2% and 4%. A defective item is found. Which shift is most likely to be responsible, and with what probability?

Q8Likely

Which of these is a partition of the sample space of the ten customer orders? (a) {Loyalty, Regular} (b) {Express, Ground} (c) {Loyalty, Express} (d) {L ∩ E, L ∩ G, R ∩ E, R ∩ G}.

Q9Likely

In a Bayes table, the prior column is 0.50, 0.30, 0.20 and the conditional column is 0.10, 0.20, 0.40. Complete the table.

Q10Likely

Fill in a probability tree for the fraud example (branches F 0.05 and L 0.95, then flagged or not flagged on each) and state which products along the branches give the joint probabilities and the total P(flagged).

Q11Possible

In your own words, what is the difference between a prior probability and a revised (posterior) probability, and what must the revised probabilities add up to?

Part 3: Discrete distributions (expected value, binomial, Poisson)

This is the second-biggest calculation block after the normal. Know the three formulas cold: the expected value μ = Σ x·P(x), the binomial P(x) = C(n, x) pˣ qⁿ⁻ˣ, and the Poisson P(x) = λˣ e⁻λ ÷ x!. Binomial and Poisson tables are not on the course scope, so questions give you numbers you can put straight into a calculator.

3A. Probability distributions and expected value

Q1Very likely

Classify each variable as discrete or continuous: (a) goals scored in a soccer game, (b) weight of a suitcase, (c) texts sent per day, (d) time taken to cook rice.

Q2Very likely

Which of these is a valid discrete probability distribution? (a) x = −1, 0, 1, 2, 3 with P = 0.1, 0.2, 0.4, 0.2, 0.1; (b) the same x with P = −0.1, 0.3, 0.4, 0.3, 0.1; (c) the same x with P = 0.1, 0.3, 0.4, 0.3, 0.1; (d) late deliveries 0, 1, 2 with P = 0.50, 0.30, 0.25.

Q3Very likely

A distribution has x = 0, 1, 2, 3, 4, 5, 6 with P(x) = 0.05, 0.12, 0.23, 0.27, ?, 0.10, 0.04. Find P(4) and then decide which statement is correct: (A) P(x > 3) = 0.67; (B) P(x < 6) = 1.00; (C) P(x ≥ 3) = 0.60; (D) P(2 ≤ x ≤ 5) = 0.21.

Q4Likely

A distribution has x = 0 to 6 with P = 0.1, 0.2, 0.1, ?, ?, 0.1, 0.1. You are told the two unknowns add to 0.4. Which can be calculated: P(X < 3), P(X ≤ 3), P(3 < X ≤ 5), P(3 ≤ X ≤ 5)?

Q5Very likely

A game pays −$1 with probability 0.3, $0 with 0.4, $3 with 0.2 and $5 with 0.1. Find the expected value and say what it means.

Q6Very likely

For the same game, find the variance and standard deviation.

Q7Very likely

A variable takes the value 0 with probability 0.1 and 2 with probability 0.9, so its mean is 1.8. If P(x = 0) rises to 0.2, what happens to the mean? A student says it stays 1.8 because 0 times anything is 0.

Q8Likely

The number of crises an executive faces on a Friday has P = 0.37, 0.31, 0.18, 0.09, 0.04, 0.01 for 0 to 5 crises. Find the mean, variance and standard deviation.

Q9Likely

Two independent sales opportunities. A pays $2,000 (probability 0.40) or $4,000 (0.60). B pays $3,000 (0.20) or $5,000 (0.80). Build the distribution of total profit and find its expected value.

Q10Likely

A store gives away a $2, $3, $4 or $5 coupon, each equally likely. Find P(X ≤ 3), the mean and the variance.

3B. Binomial distribution

Q11Very likely

Which of these is binomial, which is Poisson, and which is neither? (a) the number of heads in 10 tosses of a fair coin; (b) the number of customers arriving at a bank per 5-minute period; (c) the number of defective items in a sample of 15 when each item has a 9% chance of being defective; (d) the weight of a randomly selected package.

Q12Very likely

Three applicants are each offered a job and each accepts independently with probability 0.4. Find P(exactly 2 accept), then the mean and standard deviation of the number accepting. What are they if 300 offers are made?

Q13Very likely

In a city 10% of workers are unemployed. A random sample of 20 workers is taken. Find P(2 or fewer unemployed), P(exactly 1), and the mean and SD.

Q14Very likely

Oreos hold 10% of the cookie market. Twenty purchasers are chosen at random. Find (a) P(exactly one chooses Oreos), (b) P(fewer than four), (c) P(between two and five inclusive).

Q15Likely

8% of men are colour blind. A random sample of 10 men is taken. Find P(none colour blind), P(exactly 2), P(2 or fewer), P(at least one), and the mean and SD.

Q16Likely

65% of financial consumers are very satisfied with their primary institution. Among 25 randomly chosen consumers, find P(exactly 19 are very satisfied) and the expected number.

Q17Likely

A fair coin is flipped 10 times. Find P(exactly 3 heads) and the mean and SD of the number of heads.

Q18Possible

How does the shape of a binomial distribution change with p? Describe n = 8 for p = 0.2, 0.5 and 0.8.

3C. Poisson distribution

Q19Very likely

Customers arrive at a bank at an average of 3.2 per 5 minutes. Find (a) P(exactly 5 arrive in 5 minutes) and (b) P(exactly 10 arrive in 10 minutes).

Q20Very likely

A real estate office sells an average of 1.6 houses per weekday (Poisson). Find P(exactly 4 in a day), P(none in a day), P(more than one in a day), P(exactly 4 in two days) and P(more than 5 in a day).

Q21Very likely

Ship collisions in Vancouver harbour average 0.35 per month. Find P(no collisions in a four-month period) and P(at most one collision in four months).

Q22Likely

Customers arrive at an ice cream stand at 30 per hour. What is the probability that fewer than three arrive in the next five minutes?

Q23Likely

A clothing store averages 2.4 arrivals per 10 minutes on Saturday mornings. Find P(exactly 2 arrivals in 6 minutes).

Q24Likely

A coffee shop averages 20 customers per hour. Find P(exactly 15 customers in an hour), and the mean and standard deviation of the hourly count.

Q25Possible

A cat brings home an average of one mouse per week. Find P(exactly 4 mice in one week), and state the three conditions needed for a Poisson model.

Part 4: Continuous distributions I (uniform and exponential)

For any continuous variable a probability is an area under the curve over a range. A single exact value has probability 0, so "less than" and "less than or equal to" give the same answer here (unlike discrete variables).

Distribution	Use when	Probability	Mean and SD
Uniform on [a, b]	every value in the range equally likely	P(x₁ ≤ x ≤ x₂) = (x₂ − x₁) ÷ (b − a)	μ = (a + b)/2, σ = (b − a)/√12
Exponential, rate λ	time between events that occur at a constant average rate	P(x ≥ x₀) = e⁻λˣ⁰, P(x ≤ x₀) = 1 − e⁻λˣ⁰	μ = 1/λ and σ = 1/λ

4A. Uniform

Q1Very likely

True or false, and explain: for a continuous random variable, P(X = 3) is the height of the density curve at 3.

Q2Very likely

A bus comes every 10 minutes and you arrive at a random moment, so your waiting time is uniform from 0 to 10 minutes. Find the height of the density, P(wait between 2 and 5 minutes), and the mean and SD of the wait.

Q3Very likely

Daily gasoline sales at a station are uniform between 2,000 and 5,000 gallons. Find P(2,500 ≤ x ≤ 3,000), P(at least 4,000), and the mean and SD.

Q4Very likely

A baby's smile lasts a uniform amount of time between 0 and 23 seconds. (a) What is the probability it lasts between 2 and 18 seconds? (b) The smile has already lasted 8 seconds. What is the probability it lasts more than 12 seconds in total?

Q5Likely

Weights of machine braces in a lot are uniform from 41 to 47 grams. Find the density height, the mean and SD, P(42 ≤ x ≤ 45) and P(x > 43).

Q6Possible

A parking officer checks a 3-hour zone exactly every 3 hours. If you park and leave, on average how long until a ticket is possible, and what is the SD?

4B. Exponential

Q7Very likely

A coffee shop gets customers at a Poisson rate of 20 per hour. What distribution describes the time between consecutive arrivals, what is its rate, and what is the mean time between arrivals?

Q8Very likely

Customers arrive at a coffee shop in a Poisson process at 5 per hour. No one is in the shop now. What is the probability the barista is free for at least the next 24 minutes?

Q9Very likely

A claims counter serves 15 customers per hour (Poisson). Find P(more than 5 minutes between consecutive customers), P(less than 5 minutes) and the mean and SD of the time between customers.

Q10Likely

Bank customers arrive at a Poisson rate of 1.2 per minute. Find the mean time between arrivals and the probability that at least 2 minutes pass between arrivals. Then find P(at most 2 minutes).

Q11Likely

A battery lasts an exponentially distributed time with λ = 0.05 per hour. Find P(10 ≤ x ≤ 15) and the mean life.

Q12Likely

For any Poisson arrival process, what is the probability that the time until the next arrival is shorter than the average gap?

Q13Likely

Choose uniform, exponential, Poisson or binomial: (a) minutes until the next customer calls a help line; (b) the number of calls to the help line in an hour; (c) a random number picked anywhere between 0 and 1; (d) the number of defective bulbs in a box of 12 with a known defect rate.

Q14Possible

Describe the shape of the exponential distribution and how it changes when λ gets bigger.

Part 5: Continuous distributions II (the normal distribution)

The normal is the single most likely source of calculation marks on the exam. Use the same four steps every time: (1) sketch a bell and shade the area you want; (2) convert x to z with z = (x − μ)/σ; (3) read the table, which gives the area between the mean (z = 0) and z; (4) combine the areas. If the two x values are on opposite sides of the mean, add the areas. If they are on the same side, subtract. For a tail, take 0.5 minus the table area.

Table values used below (area from 0 to z): z = 0.44: 0.1700; 0.56: 0.2123; 1.00: 0.3413; 1.06: 0.3554; 1.25: 0.3944; 1.44: 0.4251; 1.50: 0.4332; 1.60: 0.4452; 1.94: 0.4738; 1.96: 0.4750; 2.00: 0.4772; 2.06: 0.4803; 2.50: 0.4938.

Q1Very likely

A variable has mean 50 and standard deviation 10. Convert x = 70, 60 and 40 to z-scores and interpret x = 70.

Q2Very likely

The return on an investment is normally distributed with mean 10% and standard deviation 5%. Find the probability of a loss (return below 0) and the probability the money doubles (return above 100%). Then repeat the loss probability if the SD is 10%.

Q3Very likely

GMAT scores are normal with mean 494 and SD 100. Find (a) P(494 ≤ x ≤ 600), (b) P(x > 700), (c) P(x < 550), (d) P(300 ≤ x ≤ 600), (e) P(350 ≤ x ≤ 450).

Q4Very likely

Using the standard normal table, find P(0 ≤ z ≤ 1.25), P(−1 ≤ z ≤ 0), P(−1 ≤ z ≤ 1), P(z > 1.96) and P(z < 1.25).

Q5Likely

What is the probability that a normal value lies within two standard deviations of the mean?

Q6Very likely

(a) What z-value leaves 0.4750 of area between the mean and z, so that the central 95% lies between −z and z? (b) Daily waste per person is normal with mean 2.7 kg and SD 0.78 kg. 67.72% of people produce more than what amount?

Q7Likely

Using the GMAT distribution (mean 494, SD 100), what score must you beat to be in the top 10%?

Q8Likely

Scores are normal with mean 500 and SD 50. What is the cut-off for the lowest 5%?

Q9Likely

Hotel per-diem costs are normal with SD $36. 86.65% of costs are less than $449. Find the mean.

Q10Likely

Exam marks are normal with mean 70 and SD 8. What proportion of students score above 80?

Q11Likely

A filling machine puts a mean of 500 mL in each bottle with SD 4 mL. What is the probability a bottle contains less than 490 mL?

Q12Likely

Delivery times are normal with mean 30 minutes and SD 5. Find the probability a delivery takes more than 38 minutes.

Q13Likely

Marks are normal with mean 70 and SD 10. Find P(60 < x < 85).

Q14Possible

Student A scored 82 on a test with mean 75 and SD 6. Student B scored 88 on a test with mean 80 and SD 10. Who did better relative to their class?

Q15Very likely

True or false: (a) the normal curve is symmetric about its mean; (b) its mean, median and mode are equal; (c) the curve touches the horizontal axis at ±3 SD; (d) the standard normal has mean 0 and SD 1; (e) P(x = 494) is the height of the curve at 494.

Part 6: Timed practice paper (40 multiple-choice questions)

This is a full mixed paper across all of Module 1. Treat it as a mock: close this guide's earlier parts, set a timer for 70 minutes, use a calculator and the z-table values at the top of Part 5, and write your answers on paper. Mark yourself with the key and worked solutions in Part 7. The real exam's format has not been published, so this paper tests your skill rather than predicting the layout. Aim for 34 or more correct; anything you miss, go back to the matching part and redo its questions.

Data, graphs and descriptive statistics

Q1. A survey asks customers to rate a product as Excellent, Good, Fair or Poor. The level of measurement is: (A) ordinal (B) nominal (C) interval (D) ratio.

Q2. The mean income of a random sample of 200 households is $62,000. This number is a: (A) parameter (B) population (C) statistic (D) census.

Q3. Sixty unemployment rates range from 3.6 to 12.0. You want 6 classes. The class width, rounded up to a whole number, is: (A) 1.4 (B) 1.5 (C) 8.4 (D) 2.

Q4. In a frequency table of 60 observations one class has frequency 29. Its relative frequency is: (A) 0.29 (B) 0.483 (C) 2.07 (D) 0.517.

Q5. Which graph best shows the share each of five categories contributes to a whole? (A) scatter plot (B) time-series line chart (C) pie chart (D) histogram.

Q6. The mean of 12, 15, 11, 18 and 14 is: (A) 14 (B) 13 (C) 15 (D) 70.

Q7. The median of 4, 8, 15, 16, 23, 42 is: (A) 15 (B) 15.5 (C) 16 (D) 18.

Q8. The sample standard deviation of 6, 8, 10, 12, 14 is: (A) 2.83 (B) 10 (C) 8 (D) 3.16.

Q9. A data set has mean 80 and standard deviation 12. The coefficient of variation is: (A) 6.7% (B) 12% (C) 15% (D) 96%.

Q10. A distribution has mean 52, median 47 and mode 41. It is: (A) right-skewed (B) left-skewed (C) symmetric (D) bimodal.

Q11. A value of 85 comes from a population with mean 70 and standard deviation 10. Its z-score is: (A) 0.67 (B) 1.5 (C) 15 (D) −1.5.

Q12. Using the course percentile rule, the first quartile of 106, 109, 114, 116, 121, 122, 125, 129 is: (A) 109 (B) 110 (C) 114 (D) 111.5.

Probability and Bayes

Q13. P(A) = 0.4, P(B) = 0.3 and P(A ∩ B) = 0.12. Events A and B are: (A) mutually exclusive (B) independent (C) dependent (D) complementary.

Q14. P(A) = 0.5, P(B) = 0.4 and P(A ∩ B) = 0.2. Then P(A ∪ B) is: (A) 0.9 (B) 0.2 (C) 0.7 (D) 0.1.

Q15. If P(A ∩ B) = 0.2 and P(B) = 0.4, then P(A | B) is: (A) 0.5 (B) 0.8 (C) 0.2 (D) 0.08.

Q16. At a car lot 70% of cars have air conditioning (AC), 40% have Bluetooth (BT) and 20% have both. P(BT | AC) is: (A) 0.50 (B) 0.20 (C) 0.70 (D) 0.286.

Q17. Events A and B are mutually exclusive with P(A) = 0.3 and P(B) = 0.5. P(A or B) is: (A) 0.15 (B) 0.80 (C) 0.20 (D) 0.65.

Q18. 60% of orders are placed online, and 25% of online orders are gifts. The probability that an order is both online and a gift is: (A) 0.85 (B) 0.25 (C) 0.15 (D) 0.60.

Q19. Each item tested is defective with probability 0.04, independently. P(both of two items are defective) is: (A) 0.08 (B) 0.04 (C) 0.16 (D) 0.0016.

Q20. 2% of items are defective. A test flags 90% of defective items and 5% of good items. An item is flagged. The probability it is defective is: (A) 0.269 (B) 0.90 (C) 0.018 (D) 0.049.

Q21. Line A makes 70% of output with a 1% defect rate; line B makes 30% with a 4% defect rate. A defective product is found. The probability it came from line B is: (A) 0.04 (B) 0.30 (C) 0.632 (D) 0.368.

Q22. Of 200 employees, 40 are managers and 10 of those managers are female; 70 of the 160 non-managers are female. P(female | manager) is: (A) 0.05 (B) 0.25 (C) 0.125 (D) 0.40.

Discrete distributions

Q23. Which is a valid probability distribution for x = 1, 2, 3, 4? (A) 0.2, 0.3, 0.4, 0.2 (B) 0.5, 0.6, −0.1, 0 (C) 0.3, 0.3, 0.3, 0.3 (D) 0.1, 0.2, 0.3, 0.4.

Q24. x = 0, 1, 2, 3 has P(x) = 0.1, 0.3, 0.4, 0.2. The expected value is: (A) 1.7 (B) 1.5 (C) 1.9 (D) 2.0.

Q25. The variance of the distribution in Q24 is: (A) 0.90 (B) 0.81 (C) 1.70 (D) 0.30.

Q26. A binomial experiment has n = 5 and p = 0.3. P(x = 2) is: (A) 0.031 (B) 0.132 (C) 0.309 (D) 0.400.

Q27. A binomial has n = 12 and p = 0.25. Its mean and standard deviation are: (A) 3 and 2.25 (B) 9 and 1.5 (C) 3 and 0.75 (D) 3 and 1.5.

Q28. Ten independent trials each succeed with probability 0.1. P(at least one success) is: (A) 0.651 (B) 0.349 (C) 1.000 (D) 0.100.

Q29. Calls arrive at an average of 4 per hour (Poisson). P(exactly 2 calls in an hour) is: (A) 0.073 (B) 0.147 (C) 0.195 (D) 0.238.

Q30. Customers arrive at 3 per hour (Poisson). P(no arrivals in 20 minutes) is: (A) 0.050 (B) 0.135 (C) 0.368 (D) 0.632.

Continuous distributions

Q31. A variable is uniform from 10 to 30. P(x > 22) is: (A) 0.60 (B) 0.27 (C) 0.80 (D) 0.40.

Q32. A variable is uniform from 0 to 12. Its mean and standard deviation are: (A) 6 and 3.46 (B) 6 and 12 (C) 6 and 1.73 (D) 12 and 3.46.

Q33. The time between arrivals is exponential with a mean of 5 minutes. P(more than 10 minutes) is: (A) 0.865 (B) 0.135 (C) 0.368 (D) 0.200.

Q34. Calls arrive as a Poisson process at 6 per hour. P(the next call comes within 15 minutes) is: (A) 0.223 (B) 0.250 (C) 0.777 (D) 0.632.

Q35. A normal variable has mean 100 and standard deviation 15. P(x > 115) is: (A) 0.3413 (B) 0.8413 (C) 0.0668 (D) 0.1587.

Q36. A normal variable has mean 50 and standard deviation 8. P(46 < x < 58) is: (A) 0.5328 (B) 0.4413 (C) 0.3413 (D) 0.8413.

Q37. A normal variable has mean 50 and standard deviation 10. P(x < 35) is: (A) 0.9332 (B) 0.0668 (C) 0.4332 (D) 0.1587.

Q38. Scores are normal with mean 100 and standard deviation 15. The cut-off for the top 5% is about: (A) 119.2 (B) 129.4 (C) 124.7 (D) 115.0.

Q39. Which distribution is the natural model for the time between customer arrivals? (A) Poisson (B) uniform (C) binomial (D) exponential.

Q40. A normal variable has mean 500 and standard deviation 100. P(550 < x < 700) is: (A) 0.2857 (B) 0.4772 (C) 0.6687 (D) 0.1915.

Part 7: Answer key and worked solutions for the practice paper

Answer key Open it only after you have finished the mock under timed conditions

Quick key: 1 A, 2 C, 3 D, 4 B, 5 C, 6 A, 7 B, 8 D, 9 C, 10 A, 11 B, 12 D, 13 B, 14 C, 15 A, 16 D, 17 B, 18 C, 19 D, 20 A, 21 C, 22 B, 23 D, 24 A, 25 B, 26 C, 27 D, 28 A, 29 B, 30 C, 31 D, 32 A, 33 B, 34 C, 35 D, 36 A, 37 B, 38 C, 39 D, 40 A.

Marks (out of 40)	What it means
36 to 40	Exam-ready; keep drilling the rapid review in Part 8
30 to 35	Close; redo the parts for the questions you missed
Below 30	Go back through Parts 1 to 5 in order, then retake this paper

Data and descriptive statistics

Q1 (A): the four categories have a natural order but unequal gaps, so the level is ordinal.

Q2 (C): a number computed from a sample is a statistic. A parameter describes the whole population.

Q3 (D): width = range ÷ classes = (12.0 − 3.6)/6 = 1.4, and the course rounds a class width up to a convenient whole number, 2.

Q4 (B): relative frequency = 29/60 = 0.483. Choice (D) 0.517 is the share outside the class.

Q5 (C): shares of a whole are shown with a pie chart (or a bar chart); a histogram is for a quantitative variable and a line chart is for time.

Q6 (A): (12 + 15 + 11 + 18 + 14)/5 = 70/5 = 14.

Q7 (B): n = 6, so the median is the average of the 3rd and 4th ordered values: (15 + 16)/2 = 15.5.

Q8 (D): the mean is 10, squared deviations are 16, 4, 0, 4, 16 (sum 40), and the sample variance is 40/(5 − 1) = 10, so s = √10 = 3.16. The population SD would be √(40/5) = 2.83 (choice A), but the data are a sample.

Q9 (C): CV = (12/80) × 100 = 15%.

Q10 (A): the mean is pulled above the median and the mode (mode < median < mean), which means a long right tail, so right-skewed.

Q11 (B): z = (85 − 70)/10 = 1.5.

Q12 (D): n = 8, i = 0.25 × 8 = 2, which is a whole number, so Q1 is the average of the 2nd and 3rd values: (109 + 114)/2 = 111.5.

Probability and Bayes

Q13 (B): independent events satisfy P(A ∩ B) = P(A)P(B), and 0.4 × 0.3 = 0.12.

Q14 (C): 0.5 + 0.4 − 0.2 = 0.7.

Q15 (A): P(A | B) = 0.2 ÷ 0.4 = 0.5.

Q16 (D): P(BT | AC) = P(BT ∩ AC)/P(AC) = 0.2/0.7 = 0.286. Choice (A) is the reverse conditional.

Q17 (B): mutually exclusive, so there is no overlap term: 0.3 + 0.5 = 0.8.

Q18 (C): general multiplication law with the conditional given online: 0.60 × 0.25 = 0.15.

Q19 (D): independent, so 0.04 × 0.04 = 0.0016.

Q20 (A): joints: defective and flagged 0.02 × 0.90 = 0.018; good and flagged 0.98 × 0.05 = 0.049; total flagged 0.067; P(defective | flagged) = 0.018/0.067 = 0.269.

Q21 (C): joints: A 0.70 × 0.01 = 0.007; B 0.30 × 0.04 = 0.012; total defective 0.019; P(B | defective) = 0.012/0.019 = 0.632.

Q22 (B): the given group is the 40 managers, 10 of whom are female: 10/40 = 0.25. (Choice D, 0.40, is the share of all employees who are female: 80/200.)

Discrete distributions

Q23 (D): only (D) has all probabilities between 0 and 1 and a sum of exactly 1. (A) sums to 1.1, (B) has a negative probability, (C) sums to 1.2.

Q24 (A): 0(0.1) + 1(0.3) + 2(0.4) + 3(0.2) = 0 + 0.3 + 0.8 + 0.6 = 1.7.

Q25 (B): (0 − 1.7)²(0.1) + (1 − 1.7)²(0.3) + (2 − 1.7)²(0.4) + (3 − 1.7)²(0.2) = 0.289 + 0.147 + 0.036 + 0.338 = 0.81. The SD would be 0.9.

Q26 (C): C(5, 2)(0.3)²(0.7)³ = 10 × 0.09 × 0.343 = 0.3087. Choice (A) forgets the 10 ways.

Q27 (D): μ = np = 12 × 0.25 = 3; σ = √(npq) = √(12 × 0.25 × 0.75) = √2.25 = 1.5. Choice (A) forgets the square root.

Q28 (A): at least one is the complement of none: 1 − (0.9)¹⁰ = 1 − 0.3487 = 0.651.

Q29 (B): 4² e⁻⁴ ÷ 2! = 16 × 0.01832 ÷ 2 = 0.1465. Choice (D) 0.238 is P(2 or fewer).

Q30 (C): 20 minutes is one third of an hour, so λ = 3 × 1/3 = 1 and P(0) = e⁻¹ = 0.368. Choice (A) 0.050 uses λ = 3, the wrong interval.

Continuous distributions

Q31 (D): (30 − 22)/(30 − 10) = 8/20 = 0.40.

Q32 (A): μ = (0 + 12)/2 = 6; σ = 12/√12 = 3.46.

Q33 (B): the mean is 1/λ = 5, so λ = 0.2 per minute and P(x > 10) = e⁻⁰·²×10 = e⁻² = 0.1353.

Q34 (C): λ = 6/60 = 0.1 per minute; P(x ≤ 15) = 1 − e⁻⁰·¹×15 = 1 − e⁻¹·⁵ = 1 − 0.2231 = 0.7769. Choice (A) is the "more than 15 minutes" probability.

Q35 (D): z = (115 − 100)/15 = 1.00, tail = 0.5 − 0.3413 = 0.1587.

Q36 (A): z = (46 − 50)/8 = −0.5 and z = (58 − 50)/8 = 1.0. Opposite sides, so add: 0.1915 + 0.3413 = 0.5328.

Q37 (B): z = (35 − 50)/10 = −1.5; lower tail = 0.5 − 0.4332 = 0.0668.

Q38 (C): the area from the mean to the cut-off is 0.45, so z = 1.645 and x = 100 + 1.645(15) = 124.7. Choice (B) uses z = 1.96, which is the cut-off for the top 2.5%.

Q39 (D): waiting time between arrivals is exponential; the count of arrivals would be Poisson.

Q40 (A): z = (550 − 500)/100 = 0.5 and z = (700 − 500)/100 = 2.0. Same side of the mean, so subtract: 0.4772 − 0.1915 = 0.2857.

Part 8: Rapid-fire review (20 questions in the last hour)

Cover the right-hand column, answer out loud, then check. If you hesitate on any row, go back to the part named in brackets.

#	Ask yourself	Quick answer
1	Formulas for sample variance, SD, CV and z? [Part 1B]	s² = Σ(x − x̄)² ÷ (n − 1), s = √s², CV = (σ ÷ μ) × 100%, z = (x − μ) ÷ σ. Population variance divides by N.
2	Mean or median for house prices or incomes? [Part 1B]	Median, because it is not pulled by extreme values.
3	Order of mean, median and mode when skewed right? [Part 1B]	Mode < median < mean. Skewed left reverses it.
4	Level of measurement for postal code, product rating (Excellent to Poor), temperature in °C, weight? [Part 1A]	Nominal, ordinal, interval, ratio.
5	Which statistics are meaningful for ordinal data? [Part 1A]	Mode and median. The mean is not meaningful.
6	Class width with 6 classes on a range of 8.4? [Part 1A]	8.4 ÷ 6 = 1.4, rounded up to 2.
7	Formulas for union and joint probability? [Part 2A]	P(A ∪ B) = P(A) + P(B) − P(A ∩ B). P(A ∩ B) = P(A)P(B \| A), or P(A)P(B) if independent.
8	How do you test independence? [Part 2A]	Is P(A \| B) = P(A)? Or is P(A ∩ B) = P(A)P(B)?
9	Can mutually exclusive events be independent? [Part 2A]	No, not if both have positive probability, because P(B \| A) = 0.
10	"Of those who …" or "among …" means what? [Part 2A]	A conditional probability; the group named is the "given" event in the denominator.
11	Recipe for any Bayes question? [Part 2B]	Priors × conditionals = joints. Add joints for the total. Each joint ÷ total = revised. Check revised sum to 1.
12	Expected value and variance of a discrete distribution? [Part 3]	μ = Σ x·P(x). σ² = Σ (x − μ)² P(x).
13	Binomial formula, mean and SD? [Part 3]	P(x) = C(n, x) pˣ qⁿ⁻ˣ, μ = np, σ = √(npq). Needs fixed n, two outcomes, constant p, independent trials.
14	Poisson formula, mean and variance, and the golden rule? [Part 3]	P(x) = λˣ e⁻λ ÷ x!, mean = variance = λ. Never change x; rescale λ to the interval asked about.
15	How do you do "at least one"? [Parts 2A, 3]	1 − P(none).
16	Uniform on [a, b]: probability, mean and SD? [Part 4]	P = (x₂ − x₁) ÷ (b − a), μ = (a + b)/2, σ = (b − a)/√12.
17	Exponential: tail probability, mean and SD? [Part 4]	P(x ≥ x₀) = e⁻λˣ⁰, μ = σ = 1/λ. Its λ must match the time unit of x₀.
18	Which distribution: number of arrivals versus time between arrivals? [Part 4]	Number of arrivals is Poisson; time between arrivals is exponential, with the same λ.
19	Normal procedure and the add-or-subtract rule? [Part 5]	Sketch, convert to z, read the table (area from 0 to z), then add if the x values straddle the mean, subtract if they are on the same side, and use 0.5 minus the area for a tail.
20	Reverse lookup and key z values? [Part 5]	Find the area in the table body, read z, then x = μ + zσ (z negative below the mean). Central 90% uses z = 1.645, central 95% uses z = 1.96, central 99% uses z = 2.58.

Before you start the exam, write the three formulas you worry about most on the scrap paper you are given (if allowed) and read the question wording twice, especially "at least", "more than", "between" and "given".

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