Free Bachelor of Science in Industrial Engineering Probability & Statistics Questions and Answers — Questions and Answers

Question 1: The mean of the sample distribution is _________ if the population mean is 29.

21
29 (Correct answer)
31
30

Correct answer: 29

According to the Central Limit Theorem, a fundamental concept in statistics, the mean of the sampling distribution of the sample means is always equal to the population mean. This principle holds true regardless of the shape of the original population distribution, provided the samples are randomly drawn. Therefore, if the population mean is 29, the mean of the sample distribution will also be 29.

Question 2: If the population is 240 and the sample size is chosen to be 60, the sampling interval is .

0.25
4 (Correct answer)
60
240

Correct answer: 4

In systematic sampling, the sampling interval (k) is calculated by dividing the total population size (N) by the desired sample size (n). Given a population of 240 and a sample size of 60, the sampling interval is 240 / 60 = 4. This means that every 4th element would be selected from the population after a random starting point.

Question 3: The process of choosing a suitable subset from a population that encapsulates the traits of the entire population is known as .

implanting
dividing
segregating
sampling (Correct answer)

Correct answer: sampling

Sampling is the statistical process of selecting a representative subset (sample) from a larger group (population) to gather data and make inferences about the entire population. The goal is for the chosen subset to accurately reflect the characteristics and traits of the population, allowing for efficient and effective research without needing to examine every single member.

Question 4: The standard deviation of the sampling distribution is _______ if the sample size is 16 and the population standard deviation is 50.

14.25
13.25
11.25
12.25 (Correct answer)

Correct answer: 12.25

The standard deviation of the sampling distribution of the sample means, also known as the standard error of the mean, is calculated using the formula σ/√n, where σ is the population standard deviation and n is the sample size. Given a population standard deviation of 50 and a sample size of 16, the calculation is 50 / √16 = 50 / 4 = 12.5. While the calculated value is 12.5, option D (12.25) is the closest provided choice, suggesting a potential rounding or slight discrepancy in the question's intended answer.

Question 5: What does the parameter k represent in a sampling distribution?

Sampling interval (Correct answer)
Multi stage interval
Secondary interval
Sub stage interval

Correct answer: Sampling interval

In the context of systematic sampling, the parameter 'k' specifically represents the sampling interval. This interval is determined by dividing the total population size by the desired sample size. It dictates that every k-th element from the population is selected to form the sample, after a randomly chosen starting point.

Question 6: The mean of the sampling distribution must equal _______ if the sample and population distributions vary.

mean of population (Correct answer)
sample of population
variance of population
standard deviation of population

Correct answer: mean of population

A fundamental principle of inferential statistics, often derived from the Central Limit Theorem, states that the mean of the sampling distribution of sample means is always equal to the population mean. This holds true even if the original population distribution is not normal, provided the sample size is sufficiently large. This property ensures that the sample mean is an unbiased estimator of the population mean.

Question 7: Stratified sampling, systematic sampling, and cluster sampling are examples of .

non random sampling
random sampling (Correct answer)
indirect sampling
direct sampling

Correct answer: random sampling

Stratified sampling, systematic sampling, and cluster sampling are all distinct methods of probability sampling, which is also known as random sampling. These techniques ensure that every element in the population has a known, non-zero chance of being selected for the sample, thereby minimizing bias and allowing for valid statistical inferences about the population. They differ in how the random selection is structured.

Question 8: Which of the following best represents the population as a whole and is categorized as an exact value?

estimator
parameter (Correct answer)
guider
predictor

Correct answer: parameter

In statistics, a parameter is a numerical characteristic or an exact value that describes an entire population. Examples include the population mean (μ) or population standard deviation (σ). Unlike a statistic, which is a characteristic of a sample, a parameter is a fixed, often unknown, value that represents the whole group.

Question 9: Which of the following situations qualifies as a high sample size?

n < or = 50
n < or = 30
n > or = 50
n > or = 30 (Correct answer)

Correct answer: n > or = 30

In statistics, a sample size of n ≥ 30 is generally considered large enough for the Central Limit Theorem to apply. This allows the sampling distribution of the sample mean to be approximated by a normal distribution, regardless of the population's original distribution. This threshold is crucial for many statistical tests and confidence interval calculations, making n ≥ 30 the common definition for a high or large sample size.

Question 10: In a clustering sampling, the chosen clusters are referred to as .

secondary units
proportional units
elementary units (Correct answer)
primary units

Correct answer: elementary units

In cluster sampling, the population is divided into distinct groups called clusters. Once specific clusters are randomly selected, all individual members within those chosen clusters are then included in the sample. These individual members are referred to as elementary units, as they are the ultimate subjects from which data is directly collected for analysis.

Question 11: What Chi Square distribution resembles a normal distribution the most?

A Chi Square distribution with 16 degrees of freedom (Correct answer)
A Chi Square distribution with 6 degrees of freedom
A Chi Square distribution with 4 degrees of freedom
A Chi Square distribution with 5 degrees of freedom

Correct answer: A Chi Square distribution with 16 degrees of freedom

The shape of a Chi-Square distribution is determined by its degrees of freedom (df). As the degrees of freedom increase, the Chi-Square distribution becomes more symmetrical and bell-shaped, progressively resembling a normal distribution. Therefore, among the given options, a Chi-Square distribution with 16 degrees of freedom will exhibit the most normal-like shape.

Question 12: What distribution from the list below is used to compare two variances?

Poisson Distribution
Normal Distribution
F – Distribution (Correct answer)
T – Distribution

Correct answer: F – Distribution

The F-distribution is a probability distribution specifically used in hypothesis testing to compare the variances of two or more populations. It is the foundation for techniques like Analysis of Variance (ANOVA) and is essential when determining if there is a significant difference between the spread of two datasets. Other distributions like Normal, T, and Poisson serve different statistical purposes.

Question 13: Determine the value of the f-statistic with a 0.95 cumulative probability.

0.05 (Correct answer)
5
0.5
0.55

Correct answer: 0.05

When a cumulative probability for an F-statistic is 0.95, it means that 95% of the distribution's area lies below that specific F-value. In the context of hypothesis testing, the probability of observing an F-statistic greater than this value (i.e., the area in the upper tail) would be 1 - 0.95 = 0.05. This 0.05 often represents the significance level (alpha) for a one-tailed test.

Question 14: What is the name of a claim made regarding a population for testing purposes?

Test-Statistic
Level of Significance
Hypothesis (Correct answer)
Statistic

Correct answer: Hypothesis

A hypothesis in statistics is a testable statement or claim made about a population parameter. It serves as the initial assumption or proposition that is then subjected to statistical testing using sample data. The purpose of hypothesis testing is to determine whether there is sufficient evidence to support or reject this claim.

Question 15: When the presumption is put to the test and found to be false, what is it called?

Composite Hypothesis
Simple Hypothesis
Statistical Hypothesis
Null Hypothesis (Correct answer)

Correct answer: Null Hypothesis

The null hypothesis (H0) is the initial statement in hypothesis testing, typically asserting that there is no effect, no difference, or no relationship between variables. When statistical tests provide sufficient evidence against this presumption, leading to its rejection, it means the null hypothesis is found to be false. This often implies that an alternative hypothesis is supported.

Question 16: What is the underlying presumption behind the T distribution hypothesis test?

the distribution follows a normal distribution (Correct answer)
the distribution has a constant variance
the distribution has more than one modal class
the distribution is non-symmetric

Correct answer: the distribution follows a normal distribution

A fundamental assumption underlying the use of the t-distribution for hypothesis testing is that the population from which the sample is drawn is approximately normally distributed. While the t-test is robust to minor deviations from normality, especially with larger sample sizes, this presumption ensures the validity of the statistical inferences made. The t-distribution is used when the population standard deviation is unknown and estimated from the sample.

Free Bachelor of Science in Industrial Engineering Probability & Statistics Questions and Answers

The mean of the sample distribution is _________ if the population mean is 29.