At a certain coffee shop, all the customers buy a cup of coffee; some also buy a doughnut. the shop owner believes that the number of cup s he sells each day is normally distributed with a mean of 320 cups and a standard deviation of 20 cups. he also believes that the number of doughnuts he sells each day is independent of the coffee sales and is normally distributed with a mean of 150 doughnuts and a standard deviation of 12.
a) the shop is open every day but sunday. assuming day-to-day sales are independent, what's the probability he'll sell more than 2000 cups of coffee in a week?
b) if he makes a profit of 50 cents on each cup of coffee and 40 cents on each doughnut, can he reasonably expect to have a day's profit of over $300? explain.
c) what's the probability that on any given day he 'll sell a doughnut to more than half of his coffee customers?

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

the correct answer is you spend $55.78 more in a brand store that buy ing in a normal store.

step-by-step explanation:

brand = spend   $ 135.85 total

store = spend $80.07 total

the data of the detroit pistons in numerical order from least to greatest is 0.30, 0.37, 0.55, 0.62, 0.69, 0.72, 0.73, 0.75, 0.77, 0.78, 0.80, 0.81, 0.81, 0.83, 0.84, 0.89

the data of the miami heat in numerical order from least to greatest is 0.50, 0.50, 0.56, 0.57, 0.67, 0.72, 0.72, 0.72, 0.75, 0.75, 0.76, 0.81, 0.82, 0.90, 1.00, 1.00

the detroit pistons median : 1.52

the miami heat median: 1.47

the iqr of this data is 0.05