Statistics and Probability Answers

A certain medical disease occurs in 5% of the population. A simple screening procedure is available and in 3 out of 10 cases where the patient has the disease it provides a positive result. If the patient does not have the disease there is still a 0.05 chance that the test will give a positive result. Draw a tree diagram to represent this.
a) Find the probability that a randomly selected individual
does not have the disease but gives a positive result in the screening test
gives a positive result on the test.
b) Ben has taken the test and his result is positive. Find the probability that he has the disease

The pharmaceutical company buys 400 batteries for running some specialist equipment. These batteries have a mean life of 120 hours and a standard deviation of 30 hours. Assume the battery life follows a normal distribution. [Use the Standard Normal table given in the Annexe for this task]

a) Use the standard normal table provided to calculate:
(i) the probability that a battery may work for less than 200 hours.
(ii) the probability that a battery may work for more than 200 hours.
(iii) the probability that a battery may work for less than 90 hours.

b) b) Find the number of batteries which have a life between 120 hours and 145 hours.

The company is conducting a survey to determine the spread of two strains of viruses (type A and type B) in a certain wing of a hospital. Out of 450 patients,
267 tested positive to virus A
253 tested positive to virus B
40 tested negative to both

Draw a Venn diagram to represent all the information collected in the survey.

If one patient is chosen at random, what is the probability that the patient tested is negative to both A and B?

Find the probability that a patient chosen at random:
Tested positive to virus A only.
Tested positive to virus A given that he/she tested positive to virus B.

1. Rumor has it, going to class and reading the textbook will improve your chances of doing well (i.e., better than average) in stat’s class. You notice that there are 4 students that always go to class and review the textbook everyday. To determine if the rumors are true, you ask those 4 students their marks on the quiz and find that their scores are:

60
60
61
67

Based on these scores and the knowledge that the exam scores are normally distributed (μ=50, σ=10), what can you conclude about going to every class and reviewing the textbook everyday on marks on the quiz? Do you reject or fail to reject the null hypothesis? What is the effect size? Assuming α=0.05, state your hypotheses for a one-tailed test, the critical test statistic, your conclusion, and show all your work.

A chapter of union Local 715 has 35 members. In how many different ways can the chapter select a president, a vice-president, a treasure, and a secretary?

52. According to a government study among adults in the 25- to 34-year age group, the
mean amount spent per year on reading and entertainment is $1,994. Assume that the
distribution of the amounts spent follows the normal distribution with a standard deviation
of $450.
a. What percent of the adults spend more than $2,500 per year on reading and
entertainment?
b. What percent spend between $2,500 and $3,000 per year on reading and
entertainment?
c. What percent spend less than $1,000 per year on reading and entertainment?

100 people bought tickets in a charity raffle. 60 of them bought the tickets because they supported the charity. 75 bought tickets because they liked the prize. the probability that they prize was won by someone who both supported the charity and liked the prize is 35 %. No one who neither supported nor liked the prize bought the tickets.
1. What is the probability that the prize-winning ticket was bought by someone who liked the price.
2. What is the probability that the prize was won by someone who did not support the charity.
3. What is the probability that the price was won by someone who either supported charity or liked the prize?

When a customer places an order with Candy's on-line supermarket, a computerized accounting information system automatically checks to see if the customer has exceeded his or her credit limit. Past records indicate that the probability of customers exceeding their credit limit is 0.05. Suppose that on a given day 20 customers places orders.
1. What are the mean and standard deviation of the numbers of customers exceeding their credit limits?
2. What is the probability that zero customers will exceed their limits?
3. What is the probability that one customer will exceed his or her limit?
4.What is the probability that at least two customers will exceed their limits?

7. In a study of plants, five characteristics are to be examined. If there are six recognizable differences in each of four characteristics and eight, recognizable difference in the remaining characteristics. How many plants can be distinguished by these five characteristics?
8 Let X have a uniform distribution on the interval [A,B]. compute V(X)
9 Let X have a standard gamma distribution with α=7 α=7. Compute P(X<4 or X>6)
10 Let X = the time between two successive arrivals at the drive –up window of a bank. If X has a exponential distribution with h=1 ( which is identical to a standard gamma distribution with a=1). Compute the standard deviation of the time between successive arrivals

5. A college lecturer never finishes his lecture before the end of the hour and always finishes his lecture within 2 min after the hour. Let X= the time that elapses between the end of the hour and the end of the lecture and suppose the pdf of X is
f(x)={kx 2 0 0≤x≤2otherwise f(x)={kx20≤x≤20otherwise
what is the probability that the lecture continues for at least 90 sec beyond the end of the hour?
6. Let X be a continuous rv with cdf
F(x)=⎧ ⎩ ⎨ ⎪ ⎪ 0x4 [1+ln(4x )]0 x≤05≤y≤10x>4 F(x)={0x≤0x4[1+ln(4x)]5≤y≤100x>4. What is the pdf of X?