Question

question 1:

8. Assume that the distribution of the sample in Problem 1 s N(μ, σ2). Find a 95% CI for σ2 1. The test scores in a class are 67, 83, 71, 103, 76, 95, 20, 96, 124, 79, 98, 115, 108, 113, 86, 113, 106, 83, 71, Give a 95% CI for the first quartile π1/4.

Answer 1

From the given data, the following statistics are calculated:

n = 19

$\bar{x}$ = 89.8421

s = 23.9148

$\alpha$ = 0.05

ndf = 19 - 1 = 18

Confidence interval:

$\frac{(n-1)s^{2}}{\chi _{\alpha /2}^{2}}<\sigma ^{2}<\frac{(n-1)s^{2}}{\chi _{1-\alpha /2}^{2}}$

Low End:

From Table:

$\chi _{0.025}^{2}=31.5264$

Low End:

$\frac{(n-1)s^{2}}{\chi _{\alpha /2}^{2}}=\frac{18\times 571.9181}{31.5264}=326.5367$

High End:

From Table:

$\chi _{0.975}^{2}=8.2307$

High End:

$\frac{(n-1)s^{2}}{\chi _{1-\alpha /2}^{2}}=\frac{18\times 571.9181}{8.2307}=1250.7473$

Confidence Interval:

$326.5367 < \sigma ^{2}< 1250.7473$