Statistics and Probability Answers

Q2: According to a survey by the Administrative Management Society, one-half of U.S. companies give employees 4 weeks of vacation after they have been with the company for 15 years. Find the probability that among 6 companies surveyed at random, the number that give employees 4 weeks of vacation after 15 years of employment is (a) anywhere from 2 to 5; (b) fewer than 3.

Suppose that 55% of all babies born in a particular hospital are girls. If 7 babies born in the hospital are randomly selected, what is the probability that fewer than 2 of them are girls?
Carry your intermediate computations to at least four decimal places, and round your answer to at least two decimal places.

Because of staffing decisions, managers of the Gibson-Marimont Hotel are interested in the variability in the number of rooms occupied per day during a particular season of the year. A sample of 23 days of operation shows a sample mean of 284 rooms occupied per day and a sample standard deviation of 28 rooms.
Provide a 90% confidence interval estimate of the population variance (to 1 decimal).

The probability that a patient recovers from a delicate heart operation is 0.9. What is the probability that exactly 5 of the next 7 patients having this operation survive?

A hotel managment planning to add another 700 room gotel to their chain. existing occupancy has been found to be an avarage of 70% on annual basis. it has been estimated that cist per room per annum is rs 20000. following data baswd on demand at similar hotel of tgr chain has been tabulated a) Prepare a pay off table for a xomplex with 500, 600, and 700 rooms b) Advise managment as rhe number of rooms it should construct under new propsal c) how to utilize spare capacity due to poor accupay rate.


Table

Season no of days Daily demand Avarage cost per occupied room per day


Peak Season 210 700 100

Noram Season 130 600 85

Slack Season 90 500 65

Question 3

An Italian Ice (snowcone) retailer is examining his profits for the spring season of the previous year. He realizes that his sales are based on the daily weather: sunny, average, or cold. He estimates that this year the probability of it being sunny is 0.2, average 0.5, and cold 0.3. His gross sales on these three types of days average $50, $35, and $10 respectively.

a. Find the expected income in any one day.
b. If the average supplies cost $18, what is the expected daily profit?

Juries reach judgement by unanimity rule. There is a jury of q people and 1 defendant. There are two states: J (guilty), and L (innocent). One of 2 actions is chosen: d (acquit), or f (condemn). If is state J the optimal action is f; if the state is L, the optimal action is d. No jury member has prior information, and prior probability for either state is 1/2. Each member interprets through their personal experiences, and we think of the information they receive, conditional on the state, as an independent signal. Each signal is correct with probability p > 1/2. 1. If all q members vote according to their signal, what is the probability that an innocent defendant is condemned? And the probability that a guilty defendant is acquitted? (Ruling out mistrials). 2. Imagine instead that a simple majority of "guilty" votes were sufficient for conviction. If q = 3, what is the prob. that an innocent defendant is condemned if all vote according to their signal? And the probability that a guilty defendant is acquitted?

4. A study was done to quantify the effect of cigarette smoking on standard measures of lung function in patients with idiopathic pulmonary fibrosis across different age groups. Among the measurements taken were percent predicted residual volumes. The results by smoking history were as follows
Age group Never Former Current
10-15 35 62 95
16-20 120 73 107
20-25 90 60 63
26-30 109 77 134
30-35 82 52 140
36-40 40 115 103
Above 40 68 82 158

Using the above data can we conclude that there is a difference among population residual means at 5% level of significance in terms of smoking history?

The probability of a student passing the lab test is 0.35. Two students are randomly selected to observe whether they can pass the test or not,
(i) Draw a tree diagram to illustrate the above event.
(ii) Calculate the probability that at least one person passes the test.