Name(s):                                                              :
Averages and Normality
Intro-Stat Activity
October 20, 1999
Work in pairs and one member of each team should take out a book with mostly text in it. Your statistics book is a fine example.
  1. Have one member open the book to a random page and (with eyes closed) point to a random spot on the page. If the spot (or the page itself) is not near a word in a sentence, repeat the above process until you point at a word in a sentence. Copy the sentence below, circling the word to which you pointed. If your sentence has fewer than 16 words, keep copying (the next sentence) until you record sixteen words of text.

    2.  

     
     
     
     
     
     
     
     
     
     

  2. Count the number of letters in the word you pointed to and record the value                  .

    3. Add your single word, letter count to the class data set on the board, then fill in the table below which gives the number of letters in a word, and the frequency with which that letter count occurred for the class data set.
Single word letter counts
letters in word 1  2  3  4  5  6  7  8  9  10 11 12
frequency                        
  1. Calculate the mean of the data set above for single word letter counts and the standard deviation. Make a histogram of this data set.

    2.  

     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     

  2. Now calculate the average number of letters in the first 4 words of the sentence you pointed to. The sentence you pointed to is the sentence which contains the word you pointed at. Round your average to one decimal place. Record your four word average letter count, and each individual letter count below and add your average and individual letter counts to the class data sets (stem and leaf plot and frequency table) for four word averages.
    3. Our individual letter counts
    word 1 2 3 4
    letters        

    Our average number of letters in four words                    .
     

  3. Copy the class data (stem and leaf plot) for four word letter count averages below; calculate the mean and standard deviation of the class's four word average number of letters data set. Finally record the frequencies for the class's individual letter counts in the table below (which is only needed for the last question).

    4.  

     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     

    Individual class frequencies for 4 word letter counts
    letters 1 2 3 4 5 6 7 8 9 10 11 12
    frequency                        
  4. Are the two averages (single word and four words) about the same?

    5.  

     
     
     
     
     

  5. Compare the single word standard deviation to the four word averages standard deviation. Is the four word average standard deviation about half the single word standard deviation?

    6.  

     
     
     
     

  6. Which data set is more spread out?

    7.  

     
     
     

  7. Which data set looks more normal?

    8.  

     
     
     
     

  8. Calculate the average number of letters in the first 16 words of the sentence you pointed to. If the sentence has less than 16 words, use the next several words of the second sentence until you have 16 words in all. Record your 16 word letter counts and their average (round to one decimal place) in the table below.
average = 
word 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
letters                                
  1. Add your 16 word average to the class stem and leaf plot. Copy the stem and leaf plot below.

    2.  

     
     
     
     
     
     
     
     
     
     
     
     
     

  2. Calculate the mean and standard deviation for the class data on 16 word averages.

    3.  

     
     
     
     
     

  3. Is the 16 word average mean about equal to the other means?

    4.  

     

  4. Is the 16 word standard deviation about half as large as the 4 word standard deviation?

    5.  

     
     
     
     

  5. Which of the 3 data sets is most normal looking?

    6.  

     
     
     

  6. Assuming the average number of letters in a word is a normal random variable with mean equal to the class's 16 word mean and standard deviation equal to the class's 16 word standard deviation times four, calculate the probability that a single random word has 6 or more letters in it. Compare this probability to the relative frequency of the class's single word letter counts which were 6 or more. Combine the single word letter count frequencies from questions 2 and 5, to calculate the relative frequency of words with 6 or more letters. That is, each group gave 5 observations on single word letter counts, use them all to estimate the probability that a given word has 6 or more letters in it.