Answer:
a) 0.2514
b) No
c) 0.2611
Step-by-step explanation:
For coffee cups sold daily
Mean(μ) = 320 cups
Standard deviation (σ) = 20 cups
For doughnut sold daily
Mean(μ) = 150 doughnut
Standard deviation (σ) = 12 doughnut
Let X be the random variable representing the number of cups.
Let y be the random variable representing the number of doughnuts.
a) The shop is opened everyday except Sunday's. This means that the shop is opened 6 times in a week.
The probability that he will sell more 2000 cups of coffee in a week = Pr(x>2000)
Using normal distribution
Z = (x - μ)/σ
Z = (2000 - 6(320))/6(20)
Z = (2000 - 1920) / 120
Z = 80/120
Z = 0.67
From the normal distribution table, 0.67 = 0.2486
Φ(z) = 0.2486
Recall that if Z is positive,
Pr(x>a) = 0.5 - Φ(z)
Pr(x>2000) = 0.5 - 0.2486
= 0.2514
b) On each cup of coffee, he makes a profit of 50 cents and makes a profit of 40 cents on each doughnut. We have 0.5x + 0.4y
Z = (x - μ) / σ
Z = (300 - (0.5*320 + 0.4*150)) / √0.5^2*20^2 + 0.4^2*12^2
Z = (300 - (160+60)) / √100+23.04
Z = (300 -220)/√123.04
Z = 80/11.09
= 7.02
Since 300 is more than 7 standard deviation from the mean and the value of z cannot be found in the normal distribution table, then he has NO reasonable chance to earn a profit more than 300.
c) The probability that on any given day he will sell a doughnut to more than half of his coffee customers = Pr(y - 0.5x > 0)
From normal distribution,
Z = (x - μ) / σ
Z = (0 - (150 - 0.5*320)) / √12^2 + 0.5^2*20^2
Z = (0 - (150 - 160)) / √144 + 100
Z = (0 -(-10))/√244
Z = 10/15.62
= 0.64
From the normal distribution table, 0.64 = 0.2389
Φ(z) = 0.2389
Recall that if Z is positive,
Pr(x>a) = 0.5 - Φ(z)
Pr(y - 0.5x >0) = 0.5 - 0.2389
= 0.2611
