Question

Suppose that Upper X has a discrete uniform distribution f left-parenthesis x right-parenthesis equals StartLayout left-brace1st Row 1st Column 1 divided by 3, 2nd Column x equals 1,2,3 2nd Row 1st Column 0, 2nd Column otherwise EndLayout A random sample of n equals 35 is selected from this population. Find the probability that the sample mean is greater than 2.1 but less than 2.6. Express the final answer to four decimal places (e.g. 0.9876). The probability is

Answer 1

x	f(x)	yP(x)	x2P(x)
1	1/3	0.33333	0.33333
2	1/3	0.66667	1.33333
3	1/3	1.00000	3.00000
total		2.0000	4.6667
	E(x) =μ=	ΣxP(x) =	2.0000
	E(x2) =	Σx2P(x) =	4.6667
	Var(x)=σ2 =	E(x2)-(E(x))2=	0.6667
	std deviation=	σ= √σ2 =	0.8165

since n=35 is greater than 30 , we can use normal approximation:

for normal distribution z score =(X-μ)/σ
here mean=       μ=	2
std deviation   =σ=	0.817
sample size       =n=	35
std error=σx̅=σ/√n=	0.1380

probability =P(2.1<X<2.6)=P((2.1-2)/0.138)<Z<(2.6-2)/0.138)=P(0.72<Z<4.35)=1-0.7642=0.2358