It is known that the daily revenue of the grab-and-go deli Dirk & Dijk is normally distributed.

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Question:

It is known that the daily revenue of the grab-and-go deli Dirk & Dijk is normally distributed. The manager is interested in the daily performance of her deli. A sample of revenue of 20 days is collected, and the sample mean is $320. The z-values and t-values with degree of freedom (df) equal to 19 for various confidence level (1 - )100% are provided in the table below:

Confidence Level (%) Z/2 t/2 (df = 19)

90 1.64 1.729

95 1.96 2.093

99 2.58 2.861

(i) Assuming the manager knows that the standard deviation of daily revenue equals $80, what is the 95% and 99% confidence interval of the mean revenue? How to interpret the result?

(ii) If the manager does not know the population standard deviation, she then computed the sample standard deviation to be $110. What is the 90% confidence interval of mean revenue?

Question 2:

(a) By some estimates, twenty-percent (20%) of Americans have no health insurance. Randomly sample 15 Americans. Let X denote the number in the sample with no health insurance. X has a Binomial distribution of B(15, 0.2). The cumulative Binomial distribution probabilities, i.e., P(X c) are provided below:

Cumulative Binomial probabilitues

n=15 c p = 0.2 c p = 0.2

0 0.035 8 0.999

1 0.167 9 1

2 0.398 10 1

3 0.648 11 1

4 0.836 12 1

5 0.939 13 1

6 0.982 14 1

7 0.996 15 1

(i) What is the probability that exactly 3 of the 15 sampled have no health insurance?

(ii) What is the probability that at least 1 has no health insurance?

(ii) What is the probability that fewer than 5 have no health insurance?

(b) It is Ace Heating and Air Conditioning Service finds that the amount of time a repairman needs to fix a furnace is uniformly distributed between 1.5 and four hours. Let X be the time needed to fix a furnace. Then X ~ U (1.5, 4).

(i) Find the probability that a randomly selected furnace repair requires more than three hours.

(ii) Find the 25th percentile of furnace repair times.