Stat 216 Assignment 1 - A large set of data is found to be bell-shaped

Stat 216 - Assignment #1  

For full marks you must clearly explain your answers, you will not receive any marks if only answer.

1. A large set of data is found to be bell-shaped. According to the Empirical rule, approximately 99.7% of the population (from which the data was taken) lies in the interval (60, 132). What must have been the standard deviation of the data?

Answer:

2. Consider the following probability distribution for X:

x 0 1 2 3

f(x) .1 .3 .25 .35

Calculate E(2X + 1).

Answer:

1

 

 

3. Consider the random variable X with cumulative distribution function given below. What is the probability X is an even number given that X is less than 4?

x 2 3 4 5 6

F(x) .25 .5 .6 .8 1.0

Answer:

Questions 4 and 5 refer to the following setup. A sample of 200 married men over the age of 50 were classified according to their education and number of children:

0-1 children 2-3 children Over 3 children

High school 14 37 32 Bachelors degree 19 42 17 Graduate degree 12 17 10

4. If one person is chosen at random from the sample, let E be the event that the chosen person has over 3 children, and let F be the event that the chosen person has a university degree (either bachelors or graduate). Determine which of the following statements are true and why?

(i) P (F |E) = 2759 . (ii) Events E and F are mutually exclusive.

(iii) Events E and F are independent.

Answer:

2

 

 

5. Suppose 3 people are chosen at random from the sample. Find the probability that exactly two have a high school education (ie. no university degree).

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6. For a particular experiment, if X and Y are independent events with P (X) = .7 and P (Y ) = .6, what is the probability that X or Y occurs?

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7. Celina is an executive who receives an average of 8 phone calls each afternoon between 2 and 4. Assuming that the calls are Poisson distributed, what is the probability that Celina will receive three or more calls between 2:30 and 3:30?

Answer:

3

 

 

8. After researching tumors of a particular type, a doctor learns that out of 10, 000 such tumors examined, 1500 are malignant and 8500 are benign. A diagnostic test is available which gives the correct diagnosis 80% of the time (whether the tumor is malignant or not). The doctor has discovered the same type of tumor in a new patient and decides to begin diagnosis by using the diagnostic test. If the test says that the tumor is malignant, what is the probability that the patient actually does have a malignant tumor?

Answer:

9. If X and Y are independent events with P (X) = 0.5 and P (XandY ) = 0.35, calculate P (Y ).

Answer:

10. Suppose that the probability of a bank making a mistake in processing a deposit is .002.

(a) If 5,000 deposits are audited, what is the probability that the number of mistakes that were made was no more than one? Calculate this probability directly, without using a Poisson approximation.

(b) If it is appropriate to do so, use a Poisson approximation to calculate the same probability you calculated in part (a). If it is not approriate to do so, explain why not.

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