1. 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. 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. 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:

A Bar = 8.6, B Bar = 8.2

SA-B = 1.5

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

4. 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. 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. 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. 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:

a. Assume there’s no collinearity among independent variables. Which independent variables would you recommend removing from the regression model, if any, and why?

b. How much of the variance in revenue can be attributed to the model, and how much remains unexplained by the regression model?

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?