Test 1

Answer the following three questions based on what you know about statistics now.

1. What is Statistics?

2. How is it abused?

3. How should it be used in decision-making?

4. Find the errors in the following mean, median, and mode matrix and correct them:

                                    Mean                                      Median                                   Mode

a.  5, 3, 4, 1, 2?            3                                                3                                            3

b.  5, 5, 5, 5, 5?            5                                                5                                            5

c.  3, 3, 3, 1, 3, 5?        3                                                3                                            3

d.  8, 6, 4, 5, 8, 11?   7.5                                               8                                            8

e.  1, 1, 7, 1, 1?            2.2                                             1.5                                         1

7. Calculate the variance and standard deviation of problems b-e below, using problem a as a guide:

a. 1, 2, 3, 4, 5

Step 1     Step 2    Step 3

           1     -3  =-2     x   -2 =   4

      2     -3  =-1      x  -1 =   1

      3     -3  =0      x   0 =   0

      4     -3  = 1      x   1 =   1

      5     -3  =2      x   2 =   4

Sum       15                            Sum     10 (Step 4) / (5-1) = 10/4 = 2.5 (Step 5) = V

Number    5                                                                                                                          

Average   3                                        Step 6 = the square root of V = 1.58 = SD

Now you do the following four:

b. 5, 5, 5, 5, 5

c. 1, 3, 3, 3, 3, 3, 5

d. 2, 5, 4, 5, 8, 6

e. 1, 1, 7, 1, 1

8. Using the x-axis given at the bottom of this page, construct the frequency histogram for the data set: 8, 10, 5, 11, 6, 6, 9, 12, 6, 8, 8, 20, 7, 8, 9, 7, 8, 7, 7, 13, 9, 8, 10, 8, 7, 8

5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20

Test 2. 

TOTAL  ACCIDENTS

Month                          2005               2006               2007               2008               2009

January                                               94                    59                    56

February                                             96                    80                    39

March                           33                                          127                   83

April                             53                    88                    99                    87

May                              70                    60                    83

June                             51                    83                    53

July                              90                    70                    73

August                        85                    85                    86

September                 73                    70                    70

October                       71                                            80

November                116                   70                    57

December                   83                    82                    57

1. The data period is March 2005 through April 2008 (38 months).  What is your recommendation for handling the two blank months of March and October 2006?

For the purpose of this exercise assume the two blank months are just unreported months.  Be careful to work this problem with 36 individual monthly data points.  A common error is to use 12 monthly data points.

2. The most accidents in any one month of the 36 months of data is:

3. The least number of accidents in any one month of the 36 months of data is:

4. The average number of accidents per month in the total 36 months of data is:

5. The mode of the 36 months of data is:

6. The median of the 36 months of data is: 

7. The data range of the 36 months of data (highest value - lowest value) is:

8. The variance of the 36 months of data is:

9. The standard deviation of the 36 months of data is:

Calculate these manually or with the help of a simple calculator. It is ok to use whole numbers (no decimal places) to simplify the manual calculations.

10. Sketch a run chart of the data.

11. Tally and group the data in intervals of 20 (30-49, 50-69, 70-89, 90-109, 110-129 accidents per month) and sketch a frequency histogram of the data blocks.

12. Understanding the implications of variation of data is extremely important but very often overlooked.  Consider the implications for a non-swimmer who enters a lake that is said to be three feet deep, on average.  What the non-swimmer does not know is that the lake's depth varies from 0 feet deep at the shore-line to twelve feet deep at the deepest spot near the center!)

Test 3.

1.  You have 47 employees in your department and wish to randomly select five of them to respond to a survey you have prepared.  You number the employees 01-47.  Now use the random number table on page 27 of this course material and start at the right end of the sixth row from the top of the matrix (which is number "2").  Work down the column of numbers in number pairs and select the first five random numbers you come to between 01 and 47.  What are the numbers of the five randomly selected employees you will use for the survey?

If you are having trouble with this, read on.  Here is some additional information on the Random Numbers example on the bottom of page 27 of the Course Material:

The example suggests you enter the table on the left side of the next to last row of numbers.  That would be the (horizontal) row that runs 09402 31008 53424 etc.  You need to look for pairs of numbers in the range of from 01 to 25, since you have 25 people in the population.

The first pair that works is the first two numbers: 09.  The next pair is 94.  That does not work because it is outside of the desired range.  40 is too large too but 02 works.  The next pair is 23 and that works.  Then 31-no.Then 10-ok.Then 00-no.Then 08-ok.  That gives us employees 09, 02, 23, 10, 08.  Note that the second number in a pair can be used as the first number in the second pair.

When you apply this to test three, problem one, you enter the sixth row down in the random number table.  This row starts, left to right, 31466.  Now move to the last number to the right: 2.  Now look for two digit numbers as you move down.  The first is 22.  The second is 20.  Now see if you can find the next three.

2. You time yourself driving from work to home over three weeks.  The results in minutes are as follows: 30, 35, 34, 28, 29, 33, 65, 32, 31, 30, 30, 29, 31, 30 29.  What is the:

 i. Mean driving time?

 ii. Mode?

 iii. Median?

 iv. Range?

 v.  Standard Deviation?

3. Sketch the frequency histogram of the driving times in #2.  Is the distribution normal or skewed?  Also sketch a run chart of the 15 data points.

4. Given the following data:

            data set #1: 5,5,5,5,5

            data set #2: 5,3,5,5,5

            data set #3: 4,5,3,1,2

            data set #4: 3,3,1,2,3,5,4

            data set #5: 1,3,2,4,6,7,5,9,8,

What is the range of each data set?

Which data set has the least standard deviation?

Which data set has the greatest standard deviation?

5. Construct a one-page questionnaire for twenty random employees in your organization.  Use five attributes to measure and see improved.  This questionnaire does not have to be distributed to the employees at this time.  It can be used, however, later in the course as the Term Project.  Send me a copy of this questionnaire.

Test 4

Take 10 coins, shake them in your two hands, drop them onto a table, and count and record the number of heads tossed with the 10 coins.  Don't count the tales because if you know the number of heads, you obviously know the number of tails.  Plot the number of heads tosses on a frequency histogram and run chart.  Be sure to title the two charts and label the x-axis and y-axis.  Repeat the process 20 times.

1. Is this a predictable process?  Why?

2. If we continue to gather and plot the data for 1,000 tosses what would you predict about the central tendency?  What would you predict the histogram will look like?

3. Which of the two diagrams do you like the best for plotting this data and why?

Probability = chance = odds, and is measured by: 1 in ___, % of the time, a fraction, or a decimal.

4. What is the probability of tossing "heads" in one toss of one coin?

5. What is the chance of tossing only one "heads" in two tosses of one coin?

6. What is the chance of rolling a six with one roll of one die?

Use the "Possible Combinations" chart below to help answer the following two questions.

7. What are the odds of rolling two fives with one roll of two dice?

8. What are the odds of rolling the numeric sum of "four" with one roll of two dice?

TWO  DICE

Die #1: possibilities: 1, 2, 3, 4, 5, 6

Die #2: possibilities: 1, 2, 3, 4, 5, 6

All Possible Combinations

1, 1      1, 2      1, 3      1, 4      1, 5      1, 6

2, 1      2, 2      2, 3      2, 4      2, 5      2, 6

3, 1      3, 2      3, 3      3, 4      3, 5      3, 6

4, 1      4, 2      4, 3      4, 4      4, 5      4, 6

5, 1      5, 2      5, 3      5, 4      5, 5      5, 6

6, 1      6, 2      6, 3      6, 4      6, 5      6, 6

Given a normal deck of playing cards, if one card is randomly selected, what is the probability of drawing a(n):

9. Ace?

10. Black Queen?

11. 5 of Hearts

12. Red Card

13. Black Jack of Diamonds?

14. Further develop your "10 coins" histogram and run chart, until you have a total of 50 tosses of ten coins.

15. What, if any, new ideas do you have about this process now?