Question

1.) X ~ N(60, 9). Suppose that you form random samples of 25 from this distribution. Let¯¯¯¯¯XX¯ be the random variable of averages. Let ΣX be the random variable of sums. For parts c through f, sketch the graph, shade the region, label and scale the horizontal axis for¯¯¯¯¯XX¯, and find the probability. Using Excel and Excel functions for each question show me how you got your answer.

1. ¯¯¯¯¯XX¯ ~ _____(_____,_____)
2. P(¯¯¯xx¯< 60) = _____
3. Find the 30^th percentile for the mean.
4. P(56 <¯¯¯xx¯< 62) = _____
5. P(18 <¯¯¯xx¯ < 58) = _____

Don't forget to use excel and the functions.

Answer 1

Central limit theorem:- if x follows the normal distribution with mean u and std deviation is sigma then for any n, the sample mean(xbar) follows the normal distribution with mean u & std dev (sigma/root (n))

Here X follows N(60,9) then for n =25, the sample mean xbar have mean 60 & std dev = 9 / root(25) = 9/5 = 1.8

xbar have mean 60 & std deviation 1.8

		mean		std deviation
		60		1.8
		Question	explanation	Formula	Probability
Q.1)		P(xbar < 60)		NORM.DIST(60,60,1.8,1)	0.5
Q.2)	30th Percentile	P30	P(xbar < a) = 0.30	NORM.INV(0.3,60,1.8)	59.06
Q.3)		P(56 < xbar < 62)	P(xbar < 62) - P(xbar < 56)	NORM.DIST(62,60,1.8,1) - NORM.DIST(58,60,1.8,1)	0.73
Q.4)		P(18 < xbar < 58)	P(xbar < 58) - P(xbar < 18)	NORM.DIST(58,60,1.8,1) - NORM.DIST(18,60,1.8,1)	0.13