Question

X ~ N(50, 11). Suppose that you form random samples of 25 from this distribution. Let X be the random variable of averages. Let ΣX be the random variable of sums.

A. Find the 40th percentile. (Round your answer to two decimal places.)

B. Sketch the graph, shade the region, label and scale the horizontal axis for X and find the probability. (Round your answer to four decimal places.)

P(48 < X < 54) =

C.

Sketch the graph, shade the region, label and scale the horizontal axis for X and find the probability. (Round your answer to four decimal places.)

P(17 < X < 47) =

D. Give the distribution of ΣX.

E. Find the minimum value for the upper quartile for ΣX. (Round your answer to two decimal places.)

I. Sketch the graph, shade the region, label and scale the horizontal axis for ΣX, and find the probability. (Round your answer to four decimal places.)

P(1200 < ΣX < 1350) =

Answer 1

a)

given

X ~ N[ 50 , 11 ]

sandom sample of size 25 is drawn.

Let X be the random variable of averages.

X will follow normal distribution with

mean = 50 and

standard deviation = 11 / sqrt ( 25 ) = 11 / 5 = 2.2

Hence  

X ~ N ( 50 , 2.2)

We know that X is normally distributed, with parameters:

μ=50, σ=2.2

We need to find a score x so that the corresponding cumulative normal probability is equal to 0.4. Mathematically, x is such that:

Pr(X≤x)=0.4

The corresponding z score so that the cumulative standard normal probability distribution is 0.4 is

z_c = 0.2533

This value of z_c = 0.2533

Hence, the X score associated with the 0.4 cumulative probability is

= 49.4427

b)

The following information about the mean and standard deviation has been provided:

μ=50, σ=11, n=25

We need to compute Pr(48≤Xˉ≤54).

The corresponding z-values needed to be

Therefore, the following is obtained:

c)

The following information about the mean and standard deviation has been provided:

μ=50, σ=11, n=25

We need to compute Pr(17≤Xˉ≤47).

The corresponding z-values needed to be

Therefore, the following is obtained:

d)

ΣX will also follow normal distribution

with mean = 25 * 50 = 1250

and standard deviation = 50 * 2.2 = 110

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