Q: A call center finds that 15% of calls result in a sale. If 20 calls are made, what is the expected number of sales?

$$E[X] = np = 20 \times 0.15 = 3$$

Q: A warehouse finds that 5% of orders are returned and 10% of returned orders are damaged. What is the probability that an order is returned and damaged?

$$P(Returned \cap Damaged) = 0.05 \times 0.10 = 0.005$$

Q: A call center finds that 15% of calls result in a sale. If 20 calls are made, what is the standard deviation of the number of sales?

$$\sigma = \sqrt{np(1-p)} = \sqrt{20 \times 0.15 \times 0.85} \approx 1.59$$

Question 1

A company finds that 30% of customers who receive a coupon make a purchase. If 12 customers receive coupons, what is the probability that exactly 4 customers make a purchase? Use the binomial probability formula.

Accepted Answer

$$P(X = 4) = {12 \choose 4} (0.3)^4$ (0.7)^8$

Question 2

A bank finds that 40% of customers use online banking, 30% use mobile banking, and 15% use both. What is the probability that a customer uses online or mobile banking?

Accepted Answer

$$P(Online \cup Mobile) = 0.40 + 0.30 - 0.15 = 0.55$$

Question 3

Delivery times for a shipping company are normally distributed with mean 3.5 days and standard deviation 0.8 days. What is the $$z-$$score for a delivery time of 2 days?

Accepted Answer

$$z = \frac{2 - 3.5}{0.8} = -1.875$$

Question 4

A call center finds that 15% of calls result in a sale. If 20 calls are made, what is the expected number of sales?

Accepted Answer

$$E[X] = np = 20 	imes 0.15 = 3$$

Question 5

The amount spent per online order is normally distributed with mean $$120 and standard deviation 25. $$What is the probability that a randomly selected order is more than $150?

Accepted Answer

Find $$z = \frac{150 - 120}{25} = 1.2$$, so $$P(X \u003e 150) = 1 - P(Z \u003c 1.2) \approx 1 - 0.8849 = 0.1151$$

Question 6

A warehouse finds that 5% of orders are returned and 10% of returned orders are damaged. What is the probability that an order is returned and damaged?

Accepted Answer

$$P(Returned \cap Damaged) = 0.05 	imes 0.10 = 0.005$$

Question 7

A company categorizes customers as New or Returning and whether they Made a Purchase or Did Not Purchase. What is the probability a randomly selected customer is Returning and Purchased?

Accepted Answer

$$P(Returning \cap Purchased) = 0.36$$

Question 8

A sample of 64 transactions has a population mean of 12 and standard deviation of 4. What is the standard error of the sample mean?

Accepted Answer

$$SE = \frac{4}{\sqrt{64}} = 0.5$$

Question 9

A bank finds that 15% of customers use both online and mobile banking. What is the probability a customer uses mobile banking given they use online banking?

Accepted Answer

$$P(Mobile|Online) = \frac{0.15}{0.40} = 0.375$$

Question 10

A call center finds that 15% of calls result in a sale. If 20 calls are made, what is the standard deviation of the number of sales?

Accepted Answer

$$\sigma = \sqrt{np(1-p)} = \sqrt{20 	imes 0.15 	imes 0.85} \approx 1.59$$

Question 11

The amount spent per online order is normally distributed with mean $$120 and standard deviation 25. $$What amount corresponds to the top 10% of orders?

Accepted Answer

$$X = 120 + z_{0.90} 	imes 25$$, where $$z_{0.90} \approx 1.28$$, so $$X = 120 + 1.28 	imes 25 = 152$$

Question 12

A company finds that 30% of customers who receive a coupon make a purchase. If 12 customers receive coupons, what is the expected number of customers who make a purchase?

Accepted Answer

$$E[X] = np = 12 	imes 0.3 = 3.6$$

Question 13

A warehouse finds that 5% of orders are returned. What is the probability that an order is not returned?

Accepted Answer

$$P(Not\ Returned) = 1 - 0.05 = 0.95$$

Question 14

Delivery times for a shipping company are normally distributed with mean 3.5 days and standard deviation 0.8 days. What delivery time corresponds to the slowest 5% of deliveries?

Accepted Answer

$$X = 3.5 + z_{0.95} 	imes 0.8$$, where $$z_{0.95} \approx 1.645$$, so $$X = 3.5 + 1.645 	imes 0.8 = 4.816$$

Question 15

A company categorizes customers as New or Returning and whether they Made a Purchase or Did Not Purchase. Are customer type and purchase outcome independent?

Accepted Answer

No, because $$P(Purchase|New) = 0.18/0.40 = 0.45 is $$not equal to $$P(Purchase) = 0.54$$

Question 16

A sample of 64 transactions is selected from a population with mean 12 and standard deviation 4. What is the probability the sample mean is greater than 13?

Accepted Answer

Find $$z = \frac{13 - 12}{0.5} = 2$$, so $$P(\bar{X} \u003e 13) = 1 - P(Z \u003c 2) \approx 1 - 0.9772 = 0.0228$$

Question 17

A call center finds that 15% of calls result in a sale. If 20 calls are made, what is the probability that no sales will occur?

Accepted Answer

$$P(X = 0) = (1 - 0.15$$)^{20}$$ = 0.85$$^{20}$$ \approx 0.0388$$

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