Q31: Given the eight sample values as $-4, -3, -3, 0,... - Statistical Methods & Experimental Designs - ICAR AIEEA PG Statistical Science 14 July 2023 Shift-1 Paper | Agrigyan

Calculation of student's t-test statistic:

• $2.73$ is a typical critical t-value at df = 4.
• $0.97$ is an incorrect intermediate calculation.
• $3.30$ is incorrect.
• For sample $n = 8$: $\sum X_i = -4 - 3 - 3 + 0 + 3 + 3 + 4 + 4 = 4 \implies \bar{X} = 4/8 = 0.5$. Sum of squares $\sum X_i^2 = 16 + 9 + 9 + 0 + 9 + 9 + 16 + 16 = 84$. Sample variance $s^2 = \frac{\sum X_i^2 - n\bar{X}^2}{n - 1} = \frac{84 - 8(0.25)}{7} = \frac{82}{7} \approx 11.714$. Standard error $\text{SE} = \sqrt{s^2/n} = \sqrt{11.714/8} = \sqrt{1.464} \approx 1.210$. The test statistic is $t = \frac{\bar{X} - 0}{\text{SE}} = \frac{0.5}{1.210} \approx 0.413 \approx 0.41$.