Spring 1999

May 17 class handout
 

1. Group exercises from the notes:

 
• Exercise 11.4.2
• Exercise 11.4.4
• Exercise 11.4.6

 
2. Homework due Wednesday, May 19

   (These exercises constitute a variant on Exercise 11.3.3, and they also use ideas from section 11.4.)
 
• 1. One thousand left handed students and one thousand right handed students took a test. For the left handed students, the mean score was 75, and for the right handed students the mean score was 72.
1. Assume that the standard deviation was 4 points for the left-handed students and 6 points for the right-handed students. How many left-handed students and how many right-handed students had scores:
• (i) above 80
• (ii) above 85
• (iii) above 90
• (iv) below 70
• (v) below 65
• (vi) below 60

 
2. Assume that the standard deviation was 8 points for the left-handed students and 4 points for the right-handed students. Answer the same questions as in part (a).
     Suggestions: As in exercises in the notes, you need to subtract the mean from the score in question and then divide by the standard deviation in order to get the z-value. For instance, in part (a), if we're considering left-handed students, then a score of 80 is 5 points above the mean, or 5/4=1.25 standard deviations, so that z = 1.25. Now, A(z) gives the area under the normal curve between 0 and z. We need the area under the curve which is to the right of z, so we take note of the fact that the total area under the right half of the normal curve is .5 and proceed accordingly. (Which arithmetic operation should we do?)
    
• 2. Comment on the statement that "left-handed students usually scored higher than right-handed students". In particular, does this statement apply similarly or differently under the two sets of assumptions about the standard deviations?

 
3. Some review questions

 
1. In the XYZ corporation, the salaries are as follows:

 
• 100 people earn $20,000 per year
• 20 people earn $40,000 per year
• 2 people earn $100,000 per year
• 1 person earns $300,000 per year

 
1. Determine the mean salary and the median salary

 
2. Which of these two central measures is more representative of the "average worker's" salary?

Please comment.
2. On a test , where the possible score were from 1 to 9, the following numbers of people received each score:

 

Score	0	1	2	3	4	5	6	7	8	9
#	0	0	5	10	15	25	30	20	15	5

Find the median, mean and standard deviation.
What accounts for the fact that the mean is lower than the median?
 

3. Verify the following binomial coefficient identities:

 
• 

 
• (Here we assume that k is between 1 and 100.)

 
• (Here, we assume that .)

 
• 
    (What is the correct exponent on the right side?
   
4. Verify carefully that the expected number of heads in 50 flips of a fair coin is really equal to 50.

 
5. Review your notes from class exercises, homework from chapter 11, etc.