Practice questions on Unit 1 and Unit 2
1. Select the correct interpretation of the Central Limit Theorem.
a) A population must be normally distributed for sampling to work.
b) Sample means tend to follow a normal distribution, regardless of population shape.
c) Larger samples always decrease variability.
d) The mean of any sample is equal to the population mean.
2. A fair six-sided die is rolled 10 times. What is the probability of rolling exactly four sixes?
a) 0.250
b) 0.145
c) 0.200
d) 0.089
3. A university claims that the average GPA of its students is 3.5. A researcher collects a
random sample of 50 students and finds a mean GPA of 3.4 with a standard deviation of
0.5. Which test should be used to determine if the university's claim is valid?
a) Z-test
b) T-test
c) Chi-square test
d) F-test
4. If the p-value in a hypothesis test is 0.03, what is the correct conclusion at a 5%
significance level?
a) Fail to reject the null hypothesis
b) Reject the null hypothesis
c) Increase the sample size
d) Change the significance level
5. Which of the following is NOT a valid probability value?
a) 0.75
b) -0.2
c) 1
d) 0
6. Which probability distribution is used to model the number of successes in a fixed number
of independent trials?
a) Poisson Distribution
b) Normal Distribution
c) Binomial Distribution
d) Uniform Distribution
7. A fair coin is flipped 5 times. What is the probability of getting exactly 3 heads?
a) 0.3125
b) 0.5
c) 0.1875
d) 0.25
8. The average number of calls received by a call center per minute follows a Poisson
distribution with a mean of 4. What is the probability that exactly 2 calls are received in a
minute?
a) 0.1465
b) 0.1954
c) 0.1839
d) 0.2102
9. Which of the following best defines the null hypothesis (H₀)?
a) A statement that contradicts the research question
b) A statement assumed to be true unless evidence suggests otherwise
c) A claim that must always be rejected
d) A statistical assumption that always leads to a Type I error
10. Which test is appropriate for comparing the means of two independent samples when the
population standard deviations are unknown?
a) Z-test
b) Chi-square test
c) t-test
d) F-test
11. A study finds a p-value of 0.02 in testing whether online learning improves student
performance. If the significance level is set at 0.05, what should be concluded?
a) The null hypothesis should be rejected
b) The null hypothesis should be accepted
c) The sample size should be increased
d) The test should be changed to a Z-test
12. When should a Z-test be used instead of a t-test?
a) When the population standard deviation is unknown
b) When the sample size is smaller than 30
c) When the population standard deviation is known and the sample size is large
d) When comparing more than two groups
13. The Z-test is based on which probability distribution?
a) Binomial Distribution
b) Normal Distribution
c) Poisson Distribution
d) Chi-Square Distribution
14. In a one-sample Z-test, what is the test statistic formula?
a) Z= \frac{\bar{X} - \mu}{\sigma}
b) Z= \frac{\bar{X} - \mu}{\frac{\sigma}{\sqrt{n}}}
c) Z= \frac{\bar{X} - \mu}{\frac{s}{\sqrt{n}}}
d) Z=\frac{\bar{X} - \mu}{s}
15. A researcher wants to test whether the average IQ of university students is different from
the national average of 100. A random sample of 50 students has an average IQ of 103 with
a population standard deviation of 15. Which test should be used?
a) One-sample t-test
b) Two-sample t-test
c) One-sample Z-test
d) Chi-square test
16. A company claims that the average weight of its product is 500g. A sample of 40 products
has an average weight of 495g with a standard deviation of 10g. The Z-test statistic for this
scenario is closest to:
a) -3.16
b) -2.58
c) -1.96
d) -1.28
17. A hospital claims that its average patient wait time is less than 30 minutes. A Z-test is
conducted with a p-value of 0.03 at a 5% significance level. What should be concluded?
a) Reject the null hypothesis
b) Fail to reject the null hypothesis
c) Increase the sample size
d) Use a t-test instead
18. What is the purpose of hypothesis testing?
a) To prove the null hypothesis is true
b) To make conclusions about a population based on a sample
c) To determine the probability of a Type II error
d) To find the exact value of the population parameter
19. What is the null hypothesis (H₀)?
a) A claim that there is no effect or no difference
b) A claim that always gets rejected
c) A statement that supports the alternative hypothesis
d) A conclusion based on a sample statistic
20. A p-value less than the significance level (α) means:
a) We accept the null hypothesis
b) We reject the null hypothesis
c) We increase the sample size
d) We fail to reject the null hypothesis
21. What happens when we commit a Type I error?
a) We reject a true null hypothesis
b) We fail to reject a false null hypothesis
c) We accept the alternative hypothesis
d) We decrease the significance level
22. A researcher wants to test if a new diet plan reduces weight. What is the appropriate null
hypothesis (H₀)?
a) The diet plan has no effect on weight
b) The diet plan reduces weight
c) The diet plan increases weight
d) The diet plan causes weight to remain the same
23. A university claims that the average GPA of its students is 3.2. A researcher tests this claim
using a hypothesis test and gets a p-value of 0.08 at a significance level of 0.05. What
should be the conclusion?
a) Reject the null hypothesis
b) Fail to reject the null hypothesis
c) Increase the sample size
d) Change the test to a t-test
24. A company tests if their average customer wait time is less than 10 minutes. Which test
should be used?
a) Two-tailed t-test
b) One-tailed Z-test
c) Chi-square test
d) ANOVA
25. A hypothesis test finds a p-value of 0.02. If the significance level is 0.01, what decision
should be made?
a) Reject the null hypothesis
b) Fail to reject the null hypothesis
c) Increase the sample size
d) Change the test type
26. A student is selected at random from a class. The probability that the student is a science
major is 0.4, and the probability that the student is an arts major is 0.3. If the probability
that a student is majoring in both science and arts is 0.1, what is the probability that the
student is either a science or an arts major?
a) 0.6
b) 0.7
c) 0.5
d) 0.8
27. 1.3 In a class, 60% of students like football, 50% like basketball, and 30% like both. What is
the probability that a randomly chosen student likes either football or basketball?
a) 0.8
b) 0.7
c) 0.6
d) 0.9
28. The probability that a student passes Math is 0.7, and the probability that the student
passes Science is 0.6. If passing Math and Science are independent events, what is the
probability that the student passes both subjects?
a) 0.42
b) 0.13
c) 0.58
d) 0.30
29. A company finds that the probability of a machine failing on any given day is 0.02. If two
machines operate independently, what is the probability that both will fail on the same
day?
a) 0.0004
b) 0.002
c) 0.04
d) 0.2
30. In a school, the probability that a randomly selected student plays basketball is 0.4, and
the probability that the student also plays soccer is 0.3. If 10% of students play both sports,
what is the probability that a randomly chosen student plays either basketball or soccer?
a) 0.7
b) 0.5
c) 0.6
d) 0.9
31. A factory has three machines: A, B, and C, which produce 50%, 30%, and 20% of the total
output, respectively. The probability of a defective item from machines A, B, and C is 2%,
3%, and 5%, respectively. If a randomly selected product is defective, what is the
probability that it was produced by Machine B?
a) 0.30
b) 0.35
c) 0.25
d) 0.40
32. An email filtering system detects spam emails with 95% accuracy. However, 5% of all
incoming emails are actually spam. If an email is marked as spam, what is the probability
that it is truly spam?
a) 0.50
b) 0.75
c) 0.90
d) 0.33
33. A quality control manager knows that 70% of the company’s suppliers deliver on time. If
Supplier A delivers on time 90% of the time and Supplier B delivers on time 60% of the
time, what is the probability that a shipment received on time is from Supplier A?
a) 0.65
b) 0.78
c) 0.45
d) 0.85
34. A certain university offers an admission test. 80% of students who pass the test eventually
graduate, while 40% of students who fail still manage to graduate. If 70% of students pass
the test, what is the probability that a randomly selected graduate actually passed the
admission test?
a) 0.72
b) 0.80
c) 0.55
d) 0.60
35. A police department knows that 20% of reported car thefts are false claims. If an
investigation confirms 95% of true thefts and falsely confirms 10% of false claims as true
thefts, what is the probability that a confirmed theft is actually a false claim?
a) 0.10
b) 0.20
c) 0.30
d) 0.05
36. A fair coin is flipped 10 times. What is the probability of getting exactly 6 heads?
a) \binom{10}{6} (0.5)^6 (0.5)^4
b) \binom{10}{6} (0.6)^6 (0.4)^4
c) \binom{10}{6} (0.5)^{10}
d) (0.5)^6
37. A quiz has 5 multiple-choice questions, each with 4 options. A student guesses all answers
randomly. What is the probability of getting exactly 3 correct?
a) \binom{5}{3} (0.25)^3 (0.75)^2
b) \binom{5}{3} (0.75)^3 (0.25)^2
c) (0.25)^3 (0.75)^2
d) (0.25)^3
38. A hospital receives an average of 4 emergency patients per hour. What is the probability
that exactly 2 patients arrive in an hour?
a) \frac{e^{-4}4^2}{2!}
b) \frac{e^{-2}2^4}{4!}
c) \frac{e^{-4}4!}{2^4}
d) \frac{4^2}{2!}
39. A call center receives an average of 3 calls per minute. What is the probability of receiving
no calls in a given minute?
a) 3e^{-3}
b) \frac{e^{-3}3^0}{0!}
c) \frac{e^{-3}3^1}{1!}
d) \frac{3^0}{e^3}
40. The heights of adult men in a city follow a normal distribution with a mean of 170 cm and
a standard deviation of 10 cm. What percentage of men are taller than 180 cm?
a) Approximately 16%
b) Approximately 50%
c) Approximately 34%
d) Approximately 68%
41. The weight of apples in a farm follows a normal distribution with a mean of 200 grams and
a standard deviation of 15 grams. If an apple is randomly chosen, what is the probability
that it weighs between 185g and 215g?
a) Approximately 68%
b) Approximately 34%
c) Approximately 95%
d) Approximately 50%