Test: Sampling & Estimation - UGC NET MCQ

Statement 1 is correct because the t-distribution is specifically designed for situations where the population standard deviation is unknown, particularly with smaller sample sizes (typically n < 30). This distribution accounts for the increased variability expected with smaller samples.

Statement 2 is also correct. As the sample size increases, the t-distribution becomes more similar to the normal distribution due to the central limit theorem, which states that the sampling distribution of the sample mean will approximate a normal distribution as the sample size becomes large, regardless of the shape of the population distribution.

Therefore, both statements are accurate, making the correct answer Option C: Both 1 and 2.

The standard deviation of the sampling distribution of the sample means is referred to as the standard error of the mean (SEM). It measures how much the sample means are expected to vary from the true population mean. The SEM is calculated by dividing the population standard deviation by the square root of the sample size (n). This concept is crucial in statistics because it helps to understand the precision of sample estimates. An interesting fact is that as the sample size increases, the standard error decreases, indicating that larger samples provide more accurate estimates of the population mean.

A sampling distribution is defined as the probability distribution of a particular sample statistic, such as the mean, derived from all possible samples of a given size taken from a population. This means that for each sample, the statistic is calculated, and the collection of these statistics forms the sampling distribution. This concept is crucial in inferential statistics, as it allows researchers to make conclusions about a population based on sample data. An interesting fact is that the Central Limit Theorem states that as the sample size increases, the sampling distribution of the sample mean approaches a normal distribution, regardless of the shape of the population distribution.

Both statements are correct:

1. Statement 1 is accurate because, according to the Central Limit Theorem, the sampling distribution of the sample mean will approximate a normal distribution as the sample size increases, regardless of the original population's distribution. This is true when the sample size is typically greater than or equal to 30.

2. Statement 2 is also correct because the Z-score represents the number of standard deviations a data point is from the mean. Therefore, it can be negative if the sample mean is less than the population mean and positive if it is greater. This allows for the calculation of confidence intervals above and below the mean.

Thus, the correct answer is Option C: Both 1 and 2.

The Central Limit Theorem asserts that as the sample size becomes sufficiently large, the distribution of sample means will approximate a normal distribution, regardless of the original population's distribution. This phenomenon allows statisticians to make inferences about population parameters, as the normal distribution has well-defined properties. An interesting fact is that this theorem is foundational in the field of statistics and is often used in quality control and hypothesis testing.

• The Assertion is true because a sample size of 30 is widely accepted as a benchmark for achieving a sufficiently large sample size in non-normally distributed populations.
• The Reason is also true, as the Central Limit Theorem explains why larger sample sizes result in the sampling distribution of the mean being approximately normal.
• Furthermore, the Reason provides a correct explanation for the Assertion because it justifies why a sample size of 30 is considered adequate for ensuring the reliability of statistical inferences.

- The Assertion is true. The Central Limit Theorem indeed states that the distribution of the sample means approaches a normal distribution as the sample size increases, which is a fundamental principle in statistics.

- The Reason is false. The sampling distribution of the sample means can be analyzed using parametric methods, particularly using the Z-score, which relies on the normal distribution.

- Therefore, while both statements are true, the Reason does not correctly explain the Assertion.

The mean of the sampling distribution, often referred to as the expected value, is equal to the population mean. This property is significant because it ensures that, on average, the sample means will accurately reflect the population mean, assuming random sampling. This relationship is fundamental in statistics, particularly in hypothesis testing and confidence intervals. An additional fact to note is that the variance of the sampling distribution (also known as the standard error) is equal to the population variance divided by the sample size, which illustrates how larger samples lead to more precise estimates of the population mean.

• The Assertion is true as normal distributions allow for valid inference regardless of sample size.
• The Reason is also true; however, it does not correctly explain the Assertion. While the normal distribution can accommodate various sample sizes, the statement implies that small samples are equally valid, which may not hold true for all statistical tests.
• Thus, although both statements are accurate, the Reason does not provide the correct rationale for the Assertion, making Option B the correct choice.

- The Assertion is true because point estimates indeed offer a single value that may not accurately represent the population due to variability among samples.

- The Reason is also true as interval estimates give a range of potential values, enhancing reliability compared to point estimates.

- The Reason correctly explains the Assertion, as the preference for interval estimates arises from the limitations of point estimates.