short answer of Math

1. (20 pts) A factory manufacturers machine parts. Every day, they produce 100 units, and sample 8 of

them to test for defects. One day they produce 5 defective units. What is the likelihood that they find no

defective units given that they find at most 1 defective unit?

2. (20 pts) A boutique shop models its customers’ behavior as follows: Once a customer comes into their

shop, the time until the next customer will come in follows an exponential distribution, with a mean of

10 minutes. The event that a customer entering is independent of another customer entering, and is

equally likely over the course of the day. What’s more, 2 customers will never come into the shop at the

same time.

What is the likelihood that they’ll get exactly 6 customers?

3. (20 pts) A marketing firm for a popular brand of detergent (Brand A) claim that their product is better

at removing stains than their competitor’s (Brand B). To prove this, they ask 50 people to try both

products and rate their performance. The results are as follows:

�̅� = 8.6,�̅� = 8.2

𝑠𝐴−𝐵 = 1.5

Did they prove their claim at a significance level of .05? Use a t-test.

4. (20 pts) A health food company wishes to test whether, among a certain high-risk population, their

popular vitamin drink prevents the flu. They sample 100 people within this population, whether over the

last year they a) consumed their product regularly and b) came down with the flu. They get the following

results:

48 people didn’t drink their product regularly and did not get the flu.

12 people didn’t drink their product regularly and did get the flu.

36 people drank the product regularly and didn’t get the flu.

4 people drank the product regularly and did get the flu.

Can they conclude at a significance level of .05 that there’s some relationship between use of their

product and likelihood to contract the flu?

5. (10 pts) A supermarket chain wishes to determine whether there’s a relationship between their

clients’ distance from a store and the amount they spend at checkout. They pick a sample of 10

shoppers, and ask each one how long they travelled to get there and how much they spent. They find

there is a correlation of .6325. The sample standard deviation in the amount the ten shoppers spent at

the store was $10. Can they conclude at a significance level of .05 that there’s a relationship between

shoppers’ time travelled and the amount they spend at checkout?

6. (10 pts) An insurance company provides life insurance policies to a pool of 100 high-risk clients.

Assume the time until each customer’s death to be independent random variables, and exponentially

distributed with a mean of 5 years.

To mitigate their risk, they purchase reinsurance that will kick in if 27 or more claims are filed within a

year. However, they miscalculate their reserve, and soon realize that they’ll be insolvent if 22 or more

clients file claims within the year unless their reinsurance kicks in.

Approximately what is the likelihood the company will become insolvent? (Use a normal

approximation).

7. (15 pts) A company surveys 25 of its customers, then uses the results of the survey to score them on

Income, Brand Loyalty, and Market Savvy. They take these as independent variables, and try to build

a model to predict how much their annual revenue from the customers. The output of their

regression analysis is as follows:

SUMMARY OUTPUT

Regression Statistics Multiple R 0.650 R Square 0.422 Adjusted R Square 0.335 Standard Error 127.22 Observations 24

ANOVA

df SS MS F Significance

F Regression 3 236350 78783.47 4.867513 0.010593991 Residual 20 323711 16185.57 Total 23 560062

Coefficients Standard Error t Stat P-value Lower 95% Upper 95%

Intercept 293.81 257.86 1.14 0.268 -244.07 831.70

Income 3.75 1.67 2.24 0.037 0.26 7.24

Brand Loyalty 8.08 3.00 2.69 0.014 1.81 14.34

Market Savvy -0.23 1.81 -0.13 0.901 -4.00 3.54

a. (5 pts) Assume there’s no collinearity among independent variables. Which independent

variables would you recommend removing from the regression model, if any, and why?

b. (5 pts) How much of the variance in revenue can be attributed to the model, and how much

remains unexplained by the regression model?

(5 pts) How much spending would the regression model anticipate from a customer with an income

score of 85, a brand loyalty score of 90, and a market savvy score of 40?