Question

X ~ N (60, 9). 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. To make you easy, drawing the graph, shade the region, label and scale the horizontal axis for  X− and find the probability.

X−~ N (60, 9/ 25)

9) P (X−< 60) = _____

a. 0.4

b. 0.5

c. 0.6

d. 0.7

10) Find the 30th percentile for the mean.

a. 0.56

b. 59.06

c. 5.96

d. 56.09

11) P (56 < X−< 62) = _____

a. 0.8536

b. 0.5

c. 0.1333

d. 0.56

12) P (18 < X−< 58) = _____

a. 0.8536

b. 0.5

c. 0.1333

d. 0.56

13) Find the minimum value for the upper quartile for the sum.

a. 1530.35

b. 1500

c. 1469.64

d. 66.07

14 )  P (1,400 < Σx < 1,550) = ____

a. 0.8536

b. 0.5

c. 0.1333

d. 0.6877

Answer 1

9) µ = 60, σ = 9, n = 25

P(X̅ < 60) =

= P( (X̅-μ)/(σ/√n) < (60-60)/(9/√25) )

= P(z < 0)

Using excel function:

= NORM.S.DIST(0, 1)

= 0.5

Answer B.

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10) P(x < a) = 0.3  

Z score at p = 0.3 using excel = NORM.S.INV(0.3) =    -0.5244

Value of X = µ + z*(σ/√n) = 60 + (-0.5244)*9/√25 =    59.06

Answer B)

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11) P(56 < X̅ < 62) =

= P( (56-60)/(9/√25) < (X-µ)/(σ/√n) < (62-60)/(9/√25) )

= P(-2.2222 < z < 1.1111)

= P(z < 1.1111) - P(z < -2.2222)

Using excel function:

= NORM.S.DIST(1.1111, 1) - NORM.S.DIST(-2.2222, 1)

= 0.8536

Answer A)

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12) P(18 < X̅ < 58) =

= P( (18-60)/(9/√25) < (X-µ)/(σ/√n) < (58-60)/(9/√25) )

= P(-23.3333 < z < -1.1111)

= P(z < -1.1111) - P(z < -23.3333)

Using excel function:

= NORM.S.DIST(-1.1111, 1) - NORM.S.DIST(-23.3333, 1)

= 0.1333

Answer C)

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13) mean = 60*25 = 1500

sd = 9*Sqrt(25) = 45

Distribution of ΣX ~ N(1500, 45)

P(x > a) = 0.75  

= 1 - P(x < a) = 0.75  

= P(x < a) = 0.25  

Z score at p = 0.25 using excel = NORM.S.INV(0.25) =   0.6745

Value of X = µ + z*σ = 1500 + (0.6745)*45 =    1530.35

Answer A)

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14) P(1400 < X < 1550) =

= P( (1400-1500)/45 < (X-µ)/σ < (1550-1500)/45 )

= P(-2.2222 < z < 1.1111)

= P(z < 1.1111) - P(z < -2.2222)

Using excel function:

= NORM.S.DIST(1.1111, 1) - NORM.S.DIST(-2.2222, 1)

= 0.8536

Answer A.