Question

For the formula \(t= \frac{μ_1 - μ_2}{\sqrt {\frac{S_1^2}{n_1}+\frac{S_2^2}{n_2}}}\) consider of the following statements:
A. \(μ_1\) and \(μ_2\) are sample mean and population mean respectively.
B. \(n_1\) and \(n_2\) are sample sizes of two samples from same population.
C. \(S_1\) and \(S_2\) are sample means of two samples from same population.
D. \(S_1\) and \(S_2\) are standard error of two samples from same population.
E. \(n_1\) is the sample size and \(n_2\) is the population size.
Choose the correct answer from the options given below:

• A. B and D only
• B. A, B and D only
• C. B, C and D only
• D. A and E only

Answer 1

The Correct Option is A

The given formula is used in statistics to calculate the t-statistic for two independent samples. Let's analyze each statement with respect to this formula: \(t= \frac{μ_1 - μ_2}{\sqrt {\frac{S_1^2}{n_1}+\frac{S_2^2}{n_2}}}\).

• Statement A: \(μ_1\) and \(μ_2\) are not sample means and population means. In this context, they refer to the means of two different samples.
• Statement B: \(n_1\) and \(n_2\) are indeed the sample sizes of the two samples.
• Statement C: \(S_1\) and \(S_2\) are not sample means; they represent the standard deviations or standard errors of the sample groups.
• Statement D: \(S_1\) and \(S_2\) are used to denote the standard deviations for the two samples, which are critical in calculating the denominator of this formula.
• Statement E: \(n_1\) and \(n_2\) are both sample sizes, not a mix of sample and population sizes.

Based on this analysis, the correct answer is: B and D only.