Test: Hypothesis Testing - UGC NET MCQ

The t-test calculation relies on the mean difference between the two data sets, the standard deviation of each group, and the sample size of each group. These three components are essential for determining whether the observed differences between the groups are statistically significant or merely due to chance. Understanding these factors is crucial not only for conducting a t-test but also for interpreting its results accurately. Additionally, the t-test is widely used in various fields, including psychology and medicine, to assess the effects of interventions or treatments.

- The assertion is true because the t-test is indeed used to compare the means of two sample sets.

- The reason is false because t-tests can be applied to unequal sample sizes and variances (e.g., in an unequal variance t-test).

- Since the assertion is true and the reason is false, the correct answer is Option B.

The primary purpose of conducting a t-test is to compare the average values of two data sets to see if they come from the same population. By setting up a null hypothesis that assumes the means of the two groups are equal, researchers can determine whether any observed differences in means are statistically significant or likely due to random chance. An interesting fact is that the t-test is particularly useful when sample sizes are small, making it a popular choice in experimental research, especially in fields like psychology and medicine.

- Assertion (A) is true because the Unequal Variance T-Test (Welch's t-test) is specifically designed to handle situations where the variances of the two groups being compared are not equal.

- Reason (R) is also true because the standard t-test's assumption of equal variances can distort results, leading to potentially erroneous conclusions.

- The reason provided correctly explains why the assertion is true, as it highlights the limitations of the standard t-test of variances. Thus, both statements are correct and the reason is the correct explanation of the assertion.

- Assertion (A) is true because the t-value calculation for the Unequal Variance T-Test indeed requires the means and variances of the two sample sets.

- Reason (R) is false because the degrees of freedom in an Unequal Variance T-Test are not constant; rather, they are calculated based on the sample sizes and variances, which can vary significantly between groups.

- Therefore, while the assertion is correct, the reason is not, making Option C the correct choice.

The primary purpose of a t-test is to compare the means of two groups to determine if there is a statistically significant difference between them. This method is particularly important in hypothesis testing, allowing researchers to draw conclusions about the relationship between different variables. An interesting fact about t-tests is that they can be adapted to different scenarios, such as paired t-tests for dependent samples and independent t-tests for unrelated groups, making them versatile tools in statistical analysis.

Statement 1 is correct because a z-test is indeed utilized to compare a sample mean to a hypothesized population mean when the sample size exceeds 30, and the population variance is known.

Statement 2 is incorrect because the z-statistic does not follow a t-distribution; instead, it follows a normal distribution, which is the basis of the z-test. The t-distribution is used when the sample size is small and the population standard deviation is unknown. Therefore, the correct answer is Option A: 1 Only.

The standard deviation of the differences (s(diff)) is crucial in a paired t-test as it quantifies the variability of the differences between paired observations. This measure helps in assessing how much the paired data values deviate from their mean difference, which is essential for calculating the t-value and determining if there is a statistically significant difference between the two related groups. Understanding this concept is vital in fields like psychology and medicine, where paired observations are common, such as measuring the effects of a treatment before and after its application.

- The Assertion is correct because an independent t-test indeed compares the means of two unrelated or independent groups.

- The Reason is false because, in an independent t-test, the samples should not be paired; they are independent.

- Since the Assertion is true and the Reason is false, Option C is the correct answer.

Statement 1 is incorrect because a z-test is specifically used for sample sizes of 30 or more when the population standard deviation is known. For sample sizes less than 30, a t-test is more appropriate, particularly when the population standard deviation is unknown.

Statement 2 is correct as it accurately reflects the Central Limit Theorem (CLT). The CLT states that as the sample size increases, the distribution of sample means tends to become normally distributed, regardless of the shape of the population distribution, assuming all samples are of equal size.

Thus, the correct answer is Option B: 2 Only.