Question

Suppose a simple random sample of size n = 81 is obtained from a population with p = 75 and 6 = 36. (a) Describe the sampling distribution of X. (b) What is P (X> 82.6) ? (c) What is P (xs 66.8) ? (d) What is P (69.2<x< 82.8) ?

Answer 1

a) according to central limit theorem sampling distribution of sample mean is normally distributed with mean 75 and standard deviation 36/√81 = 36/9 = 4

b)

The following information about the mean and standard deviation has been provided: M = 75, o = 36, n = 81 We need to compute Pr(X > 82.6). The corresponding z-value needed to be computed is: Í - M Z 82.6 – 75 36/V81 1.9 0/vn Therefore, we get that 82.6 – 75 Pr( X > 82.6) = Pr (Z > (z Pr(Z > 1.9) 36/V81 =1 – 0.9713 0.0287 The following is obtained graphically: Normal Distribution: Pr(Xbar>82.6)=0.0287 0.100 0.090 0.080 0.070 0.060 0.050 0.040 0.030 0.020 0.010 0.000 58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92

c)

The following information about the mean and standard deviation has been provided: A = 75, 0 = 36, m = 81 We need to compute Pr(X < 66.8). The corresponding z-value needed to be computed: X - M Z 66.8 – 75 36/V81 = = -2.05 o/vn Therefore, 66.8 – 75 Pr( X < 66.8) = Pr (Z< 36/V81 = Pr(Z < -2.05) = 0.0202 The following is obtained graphically: Normal Distribution: Pr(Xbar<66.8)=0.0202 0.100 0.090 0.080 0.070 0.060 0.050 0.040 0.030 0.020 0.010 0.000 58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92

d)

The following information about the mean and standard deviation has been provided: A = 75, 0 = 36, m = 81 We need to compute Pr(69.2 < X < 82.8). The corresponding z-values needed to be computed are: X1 - 21 = 69.2 - 75 36/781 = -1.45 oln 22 X2 – U o/vn 82.8 – 75 36/V81 = 1.95 Therefore, the following is obtained: 69.2 - 75 Pr(69.2 < X < 82.8) = Pr <2< 82.8 – 75 36/781 = Pr(-1.45 < 3 < 1.95) 36/V81 - = Pr(Z < 1.95) – Pr(Z < -1.45) = 0.9744 0.0735 = 0.9009 The following is obtained graphically: Normal Distribution: Pr(69.2<Xbar<82.8)=0.9009 0.100 0.090 0.080 0.070 0.060 0.050 0.040 0.030 0.020 0.010 0.000 58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92

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