I. The standard error will decrease as the sample size increases.
II. The standard error will increase as the sample size increases.
III. The mean equals the population mean.
IV. The mean is not equal to the population mean.
Analyzing the Properties of a Random Sampling Distribution of Means
Understanding the Sampling Distribution of Means
A random sampling distribution of means is created by taking numerous random samples of a specific size from a population and calculating the mean for each sample. The distribution of these sample means has specific, predictable characteristics.
Evaluating Statement I & II: Standard Error and Sample Size
The standard error (SE) of the sampling distribution of means measures the variability of sample means around the population mean. It is calculated using the formula:
$SE = \frac{\sigma}{\sqrt{n}}$
where $\sigma$ is the population standard deviation and $n$ is the sample size.
- Statement I Analysis: As the sample size ($n$) increases, the denominator ($\sqrt{n}$) also increases. This leads to a decrease in the value of the standard error ($SE$). Therefore, Statement I is true.
- Statement II Analysis: Based on the relationship described above, the standard error decreases, not increases, as the sample size increases. Therefore, Statement II is false.
Evaluating Statement III & IV: Mean of the Distribution
A key property of the random sampling distribution of means is its central tendency.
- Statement III Analysis: The mean of the sampling distribution of means ($\mu_{\bar{x}}$) is equal to the population mean ($\mu$). This is a fundamental principle known as the Law of Large Numbers in this context. So, $\mu_{\bar{x}} = \mu$. Statement III is true.
- Statement IV Analysis: Since the mean of the sampling distribution equals the population mean, Statement IV is false.
Conclusion
Based on the analysis, statements I and III accurately describe the properties of a random sampling distribution of means.
- Statement I: True (SE decreases as n increases)
- Statement III: True (Mean of the distribution equals population mean)
Therefore, the correct option combines statements I and III.
Important Questions from Sample - Teaching
- Which of these statements does NOT describe snowball sampling correctly?
A. It is a non-probability sampling technique.
B. It is used when the target population is readily available.
C. Initial participants refer others to take part in the study.
D. It ensures equal chance of selection for all individuals.
E. It is commonly used in sociological and qualitative research.
Choose the correct answer from the options given below: - Assertion (A): The larger the sample size, the greater will be the power of the test.
Reason (R) : A large sample size results in smaller standard error of mean.
Codes : - Given below are two statements, one labelled as Assertion (A) and the other labelled as Reason (R). Read the statements and choose the correct answer using the code given below.
Assertion (A) : Findings of a research based on a large sample size of a population being studied can always be generalized to the whole population.
Reason (R) : Sample size does not ensure representativeness. In a study, participants are recruited from local schools in urban area. The study is conducted on saturday so only the saturday working school take part. What type of sample is this ?
Multistage cluster sampling is helpful when :
