Suppose that the weights (in kgs) of six months old babies, monitored at a healthcare facility, have
N ( μ , σ 2 ) N(\mu, \sigma^2) N ( μ , σ 2 ) distribution, where
μ ∈ R \mu \in \mathbb{R} μ ∈ R and
σ > 0 \sigma > 0 σ > 0 are unknown parameters. Let
X 1 , X 2 , … , X 9 X_1, X_2, \ldots, X_9 X 1 , X 2 , … , X 9 be a random sample of the weights of such babies. Let
X ‾ = 1 9 ∑ i = 1 9 X i \overline{X} = \frac{1}{9} \sum_{i=1}^{9} X_i X = 9 1 ∑ i = 1 9 X i ,
S = 1 8 ∑ i = 1 9 ( X i − X ‾ ) 2 S = \sqrt{\frac{1}{8} \sum_{i=1}^{9} (X_i - \overline{X})^2} S = 8 1 ∑ i = 1 9 ( X i − X ) 2 and let a 95% confidence interval for
μ \mu μ based on
t t t -distribution be of the form
( X ‾ − h ( S ) , X ‾ + h ( S ) ) (\overline{X} - h(S), \overline{X} + h(S)) ( X − h ( S ) , X + h ( S )) , for an appropriate function
h h h of random variable
S S S . If the observed values of
X ‾ \overline{X} X and
S 2 S^2 S 2 are 9 and 9.5, respectively, then the width of the confidence interval is equal to __________ (round off to 2 decimal places) (You may use
t 9 , 0.025 = 2.262 , t 8 , 0.025 = 2.306 , t 9 , 0.05 = 1.833 , t 8 , 0.05 = 1.86 t_{9,0.025} = 2.262, t_{8,0.025} = 2.306, t_{9,0.05} = 1.833, t_{8,0.05} = 1.86 t 9 , 0.025 = 2.262 , t 8 , 0.025 = 2.306 , t 9 , 0.05 = 1.833 , t 8 , 0.05 = 1.86 ).