Question

The weekly wages of 1000 workmen are normally diturbed with a mean of 2000 and a standard deviation of 150.Estimate the number of workers whose wages lie between 1750 and 2250?

Answer 1

Answer

z-tables have mean value = 0 and probability at P(z<= 0) = 0.50

ZTable(z < 1.0) gives the probability for z to be between 0 and 1.0 = P(0 < z <1.0)

So, P(z<1.0) = P(z<0) + P(0<z<1.0) = 0.50 + ZTable(1.00)

Also, P(z< -2.0) = P(z<0) - P(-2<Z<0) = 0.5 - P(0<Z<2.0) = 0.5 - ZTable(2.00)

P(3<z<4) = ZTable (4.00) - ZTable(3.00)

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Mu = mean = average = Rs 70, N = 2000, Sigma = Rs 5 = std deviation

X = wage Z = normal distribution function variable = [ X - Mean ] / Sigma

1) Number of estimated workers with Wages X between Rs 70 and Rs 71

X2 = Rs 71 Z2 = (71-70)/5 = 0.2

X1 = Rs 70 Z1 = (70-70)/5 = 0

P( 70 < X < 71) = P(0<Z<0.20) = ZTable (0.20) = 0.0793

Estimation of Workers : N * Probability = 2000 * 0.0793 = 158.6 or 159

2) Estimation of workers with wages Rs 69 < X < Rs 73

Z2 = (73-70)/5 = 0.6 Z1 = (69-70)/5 = -0.2

P(-0.2 < Z < 0.6) = P(-0.2 <Z<0) + P(0<Z<0.6) = P(0<Z<0.2) + P(0<Z<0.6)

= ZTable(0.2)+ZTable(0.6) = 0.0793 + 0.2257 =0.3050

Estimation of Workers : N * Probability = 2000 * 0.3050 = 610

3) Estimation of workers with wages more than Rs 72

Z = (72-70)/5 = 0.4

P(Z>0.4) = 1 - P(Z<0.4) = 1 - [ P(Z<0) + P(0<Z<0.4) ] = 1 - [ 0.5 + ZTable(0.4)]

= 0.5 - ZTable (0.4) = 0.5 - 0.1554 = 0.3446

Estimation of Workers : N * Probability = 2000 * 0.3446 = 689.2

4) Estimation of workers with wages less than Rs 65

Z = (65-70)/5 = -1

P(Z< -1) = P(Z<0) - P(-1<Z<0) = 0.50 - P(0<Z<1) = 0.50 - ZTable(1.00)

= 0.500 - 0.3413 = 0.1587

Estimation of number of workers : N * Probability = 317.4

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If you have error function values, then Multiply the following probabilities with N =2000.

a) 1/2 { erf [ (71-70)/ 5 √2 ] } = 1/2 erf (1/5√2)

b) 1/2 { erf (73 - 70) / 5√2 ] - erf [(69 - 70)/5√2 ] } = 1/2 [ erf (3/5√2) - erf (-1/5√2) ]

= 1/2 [ erf (3/5√2) - 1 + erf(1/5√2) ]

c) 1/2 { 1 - erf [ ( 72 - 70)/ 5√2 ] } = 1/2 [ erf (√2 * 1/5 )

d) 1/2 { 1 - erf [ 5 / 5√2] } = 1/2 { 1 - erf (1/√2) ]

MP