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The mass of the eggs laid by a certain breed of hen is a normally distributed random variable with mean 0.06kg and standard deviation 0.009kg. If a random sample of 1800 eggs are sold, how many would you expect to have mass; (i)Less than 0.052 kg (standard size). (ii)Between 0.052 kg and 0.075kg (medium size).(iii)Greater than 0.075 kg (large size).

(b)If the cost of producing an egg in (a) is Gh¢0.45 and selling price of each standard, medium and large size eggs are Gh¢0.60, Gh¢0.80, and Gh¢0.95 respectively, calculate (i)The expected profit to be made from selling the standard size, medium size and large size eggs. (ii)The overall profit.​

Respuesta :

onyebuchinnaji

a (i) The number of standard size is 288 eggs

a (ii) The number of medium size is 1,476 eggs

a (iii) The number of large size is 36 eggs

b (i) The expected profit made from selling the standard size is Gh¢43.2

b (ii) The expected profit made from selling the medium size is Gh¢516.6

b (iii) The expected profit made from selling the large size is Gh¢16.2

The given parameters:

the mean of the distribution, m = 0.06 kg

standard deviation (std) of the distribution, d = 0.009 kg

number of the samples, n = 1800 eggs

(i) Find the position of "less than 0.052 kg (standard size)":

  • 1 standard deviation below the mean = m - d
  • m - d = 0.06 kg  -  0.009 kg = 0.051 kg

(the standard size is 1 standard deviation below the mean)

  • less than 1 standard deviation below the mean in a normal distribution is equal to 16% of the data samples
  • Number of standard size = 0.16 x 1800 = 288 eggs

(ii) Find the position of "Between 0.052 kg and 0.075kg (medium size)":

  • 0.052 kg is 1 standard deviation below the mean
  • 2 standard deviation above the mean = m + 2d
  • m + 2d = 0.06 + 2(0.009) = 0.078 kg

(the medium size is between 1 std below the mean and 2 std above the mean)

  • Between 1 std below the mean and 2 std above the mean in a normal distribution = (68 + 14)% = 82%
  • Number of medium size = 0.82 x 1800 = 1,476 eggs

(iii) Find the position of "Greater than 0.075 kg (large size)":

  • 0.078 kg is 2 standard deviation below the mean
  • greater than 2 std above the mean in a normal distribution = 2 % of the data samples
  • Number of large size = 0.02 x 1800 = 36 eggs

Check: 288 eggs + 1476 eggs + 36 eggs = 1,800 eggs

(b) the cost of production of an egg = Gh¢0.45

the selling price of the standard size = Gh¢0.60

the selling price of the medium size = Gh¢0.80

the selling price of the large size = Gh¢0.95

(i) The expected profit made from selling the standard size:

Profit = total revenue - cost of production

Profit = 288(0.6) - 288(0.45) = Gh¢43.2

(ii) The expected profit made from selling the medium size:

Profit = total revenue - cost of production

Profit = 1476(0.8) - 1476(0.45) = Gh¢516.6

(iii) The expected profit made from selling the large size:

Profit = total revenue - cost of production

Profit = 36(0.9) - 36(0.45) = Gh¢16.2

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