Practice Exam Môn Engineering Probability and Statistics | Trường Đại học Quốc tế, Đại học Quốc gia Thành phố Hồ Chí Minh

lOMoAR cPSD| 58583460 Practice Exam
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SECTION 1: Problem-Solving Questions
INSTRUCTION: To get full marks, you need to write down the basic steps to obtain your final answers. Apply
3-digit to your answered numbers.
Question 1: Given the following sample data set. 14 12 16 13 11 12 17 14
a. Find the sample mean, the sample median and the sample mode.
b. Find the sample variance and the 65% percentile (10 pts).
Now assume that this sample is from an IID random sample. One wants to learn about the unknown
mean 𝜇 of the population that is assumed to generate the data by computing the confidence interval for the data.
c. State the necessary assumptions, and find the 95% two-sided confidence interval.
d. If want to reduce to upper and low values for the 95% CIs to 𝑥̅ ± 1. What is the minimum sample
size? Assume that the sample standard deviation doesn’t change.
Note: Be precise on the notations you use to get full marks.
Question 2: A medical scientist believes that the average temperature of healthy individuals has increased
over time and is now greater than 37 degrees Celsius. To prove this, she has randomly selected 100 healthy
individuals. It is found that their sample mean temperature is 37.09 with a sample standard deviation of 0.4 degrees.
a. State the two-sided hypotheses for the test.
b. Perform the test to evaluate the claim at 5% significance level.
c. Perform a one-sided test (use 𝐻0 with ≥) to evaluate the claim at 5% significance level.
d. Compute the 𝑝-value for the one-sided test in (c). Without using the test statistic and the rejection
region, utilize the 𝑝-value at evaluate the claim at 1% significance level. Justify your choice.
Question 3: An economist has a 200-observation dataset including the average wage (thousand VND per
hour) and the working experience (the number of years).
a. The dataset includes wage earners from three cities: Ha Noi, Da Nang and Ho Chi Minh city. He
performs an ANOVA analysis to check equality of the mean from the three provinces, and obtain a table as: Sum of squares Mean Square The test statistic 𝑆𝑆𝑏 = 2,531.235 𝑀𝑆𝑏 = Ftest = 𝑆𝑆𝑊 = 147,382.354 𝑀𝑆𝑊 =
State the 𝐻0 and 𝐻1, fill in the table and test the hypothesis test at 5% significance level. It is known
that the F critical value is 2.650. lOMoAR cPSD| 58583460
b. A regression model is used to learn how wages can be explained by working experiences. Write
down the regression model the economist can use? Note: you need to be accurate on what are
the dependent and independent variables.
c. The estimated results show that 𝛽̂ 1 = 2.150 and 𝑠𝑒(𝛽̂ 1) = 0.350. State the hypothesis and perform
two-sided hypothesis testing for 𝛽̂ 1 being equal to zero or not. Compute the associated 𝑝-value of the test.
Note: Be precise on the notations you use to get full marks.
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SECTION 2: Multiple-choice Questions INSTRUCTIONS: There
is only one correct answer for each question.
Question 1: The following is a histogram of ages of people applying for a job position.
Which of the following statements are true?
I. The median age is between 24 and 25. II.
The mean age is between 22 and 23.
III. The mean age is greater than the median age. A. I only B. II only C. III only D. All are true.
Question 2: The Central Limit Theorem states that the sampling distribution of the sample mean is
approximately normal under certain conditions. Which of the following is a necessary condition for the
Central Limit Theorem to be used?
A. The sample size must be large (e.g., at least 30).
B. The population size must be large (e.g., at least 30).
C. The population from which we are sampling must be normally distributed.
D. The population from which we are sampling must not be normally distributed.
Question 3: We have a population distribution with a mean and a variance. Then, a sample is collected.
What are the correct notations, we use in this course, for ‘the standard variation’ and ‘the computed value of the sample mean’.? A. 𝜎 and 𝑋̅ . lOMoAR cPSD| 58583460 B. S and 𝑋̅ . C. 𝜎 and 𝑥̅ . D. S and 𝑥̅ .
Question 4: A piece of medical equipment is not functioning properly, however, in running operational
checks, a lab technician does not find evidence of the malfunction. The lab technician has committed an equivalence of: A. a Type I error. B. a Type II error.
C. both a Type I and a Type II error.
D. neither a Type I nor a Type II error.
Question 5: Comparing between a 95% z-confidence interval to a 95% t-confidence interval, which of the
following statements is INCORRECT?
A. The z-confidence interval would be wider than the t-confidence interval.
B. The z-confidence interval would be narrower than the t-confidence interval.
C. Both intervals are nearly identical if the sample size is large, e.g. 𝑛 ≥ 100.
D. They need to use the assumption of a random sample from a normal population.
Question 6: After collecting a sample, a factory finds the 95% confidence interval of the mean length of a
produced item is 10.01 ± 0.05 cm. The manager wants to decrease the margin of error from ±0.05 to
±0.01 (to get a value such as 10.01 ± 0.01), how the manager can do it?
A. The sample size must increase by a factor of 5.
B. The sample size must increase by a factor of 25. C. Nothing can be done.
D. Not enough information is given.
Question 7: The hypothesis testing, when the p-value of another new data is smaller than the current
data’s, which of the following statements is true?
A. The stronger the evidence against the null hypothesis provided by the current data.
B. The weaker the evidence against the null hypothesis provided by the current data.
C. The evidence provided by the new data is more extreme than the one by the current data.
D. The evidence provided by the new data is less extreme than the one by the current data.
Question 8: Which of the following assumptions does NOT apply to ANOVA? A. The sample is random.
B. Normal distribution for all groups with possibly different of the mean of each group.
C. The variances of each group are identical.
D. The sample size is equal among groups. lOMoAR cPSD| 58583460 lOMoAR cPSD| 58583460 lOMoAR cPSD| 58583460