Question

Data

For Tasks 1-8, consider the following data:

7.2, 1.2, 1.8, 2.8, 18, -1.9, -0.1, -1.5, 13.0, 3.2, -1.1, 7.0, 0.5, 3.9, 2.1, 4.1, 6.5

In Tasks 1-8 you are asked to conduct some computations regarding this data. The computation should be carried out manually. All the steps that go into the computation should be presented and explained. (You may use R in order to verify your computation, but not as a substitute for conducting the manual computations.)

A Random Variable

In Tasks 9-18 you are asked to conduct some computations regarding a random variable. Use the (incomplete) table below as the definition of this random variable (after you fill in the blank). The sample space of a random variable is comprised of the integers 0, 1, 2, 3, 4, 5, and 6. The probabilities of each value are shown in the table below (with one missing value).

Value

0

1

2

3

4

5

6

Probability

.10

.15

.25

.10

.10

.15

A Population

For Tasks 19-21, use the file called "pop3.csv" found here. That file contains information about time to failure of an entire production of some computer parts. The file contains two variables, "type" and "time", each measured over the 100,000 members of the population. The variable "type" is a factor, with three levels, "a", "b" and "c", and the variable "time" is numeric. If the value of time is 4, that means that the part lasted 4 units of time (years?) before a failure occurred. You should treat the content of this file as the information from an entire population.

Save the file on your computer and read the data stored in the file into R. If you have trouble loading the data into R, email your instructor immediately—don't worry if you think you will find the answer 1 minute after sending the email—just send the email. Tasks 19-21 refer to the information in the file.

Submitting the Assignment

For the assignment you should complete the following tasks. Tasks 1-8 refer to the sequence of 17 data values presented above, Tasks 9-18 refer to the random variable and Tasks 19-XXX21 refer to the information of a population of computer parts that is stored in the file "pop3.csv". Your answers should be short and clear.

We recommend that you copy and paste the tasks below into the assignment submission area. You can then write your answers to the tasks in the designated positions that are marked in the text:

Tasks

Data:

1. Using the list of 17 numbers at the top of the page, the median of this data, rounded to two decimal places, is:_____.

2. If you find the median using the original method (paper and pencil), you have to arrange the values into numeric order (True/False).______________________

3. The interquartile range for this data is (round each value to 3 decimal places):_______.

4. The formula for calculating the interquartile range is_____________ (show the formula and a citation to the source that you used).

5. Using techniques that we have studied in this course, the upper and lower cutoff points (rounded to three decimal places) for identifying outliers (in the given data sample) are: ________ and ________ (this is not a request to show any outliers--just the cutoff points that would determine what constitutes an outlier). You may round to three decimal places.

6. The summary() command shows a list of outliers, if there are any (True/False):______________________

7. The list of outlier values is:_____________ (if there are none, write "NA").

8. The standard deviation of the list of 17 numbers is (round to 3 decimal places): ______________

A Random Variable:

9. The missing probability value (under the number 4) in the random variable table above is:_______

10. The sum of the probabilites in the second row of any random variable table like the one above should equal (round to 3 decimal places): _______________________________________________.

11. Read section 4.4.1 in the book (Yakir, 2011). Do the numbers in the table above (for the random variable) represent a data sample (Yes/No)?____

12. In the random variable table shown above, the value in the second row represents the cumulative probability of the corresponding values in the first row (True/False) _________

13. The probability that a randomly selected value from this random value will be less than or equal to 3 is :_____.

14. What is the probability that a randomly selected value from the random variable would be exactly 1.5? ___________ .

15. Review section 4.4 in the book (Yakir, 2011), especially pages 57—58. The expectation of the random variable is:______.

16. To find the expectation of a random variable by using a relative frequency table, you can add the values in the first row of the table and divide by the number of columns in the table (True/False)_________.

17. Study Yakir (2011) pp. 57-59 and solved problems 4.1.6-4.1.8. The (population) standard deviation of the random variable above is (round to 3 decimal places):_______ (hint, you can not put values from the table into the sd() function because the sd() function does not adjust for the probabilities).

18. If you have already calculated the standard deviation of a data sample, what is the next thing to do to find the variance: ______________________________.

A Population:

19. Determine how many observations in the pop3.csv file are of type a: _______.

20. Using the appropriate R function with the defaul options, what is the median of the time column of pop3 (round to 3 decimal places): ______________________________.

21. What is the variance of the time column of pop3 (rounded to three decimal places)? _______

Answer 1

Data is :

7.2, 1.2, 1.8, 2.8, 18, -1.9, -0.1, -1.5, 13.0, 3.2, -1.1, 7.0, 0.5, 3.9, 2.1, 4.1, 6.5

The data arranged inascending order is

-1.9,-1.5,-1.1,-0.1,0.5,1.2,1.8,2.1,2.8,3.2,3.9,4.1,6.5,7.0,7.2,13.0,18

Answers :

1. Using the list of 17 numbers at the top of the page, the median of this data, rounded to two decimal places, is: 2.8

Solution : Md = (n+1)/2 = 9th observation of the arranged data

Median = 2.8

2. If you find the median using the original method (paper and pencil), you have to arrange the values into numeric order : True

3) The interquartile range for this data is (round each value to 3 decimal places) : 6.550

First Quartile = (n+1)/4 = 4.5th observation

Q1 =-0.1 +0.5(0.5-(-0.1) = 0.2

Thrid Quartile = 3(n+1)/4 = 13.5 th observation

Q3 = 6.5+0.5(7-6.5) = 6.75

Inter Quartile Range = Q3-Q1 = 6.75- 0.2 = 6.550

Q4 ) The formula for calculating the interquartile range is IQR = Q3-Q1

Q5) Upper Cut Off = Q3+1.5 IQR =6.75+1.5*6.55 = 16.575

Lower Cut Off = Q1- 1.5IQR = 0.2-1.5*6.55 =-9.625

6) True

7) The outlier is 18 since it is greater than Upper Cut off 16.575

8) The Standard deviation of the list of numbers is