Video Transcript
In an extract of a book, the number
of words per sentence was counted. Find the missing numbers in the
following table. Use the previous table to calculate
an estimate for the mean number of words. Give your answer to two decimal
places.
In this question, we are given a
grouped frequency table that lists the number of words per sentence in a book along
with their frequencies. 15 sentences had between one and
seven words inclusive. 20 sentences had between eight and
14 words. 45 sentences had between 15 and 21
words. And we also have the frequencies
for between 22 and 28 words, 29 and 35 words, and 36 and 42 words. The third column of our table
corresponds to the midpoints. To calculate these values, we add
the two boundary values in each group and divide by two. For example, in the first row, we
add one and seven to give us eight, and dividing this by two gives us four. This is the midpoint of the group
one to seven.
We need to calculate the midpoint
for the second group in our table, where the number of words are between eight and
14 inclusive. The midpoint is therefore equal to
eight plus 14 divided by two. This simplifies to 22 divided by
two, which is equal to 11. The final column of our table
corresponds to the product of the frequency and the midpoint. In the first row, 15 multiplied by
four is equal to 60. We need to calculate the frequency
of 20 multiplied by the midpoint of 11. And this is equal to 220. The two missing numbers in the
table are 11 and 220.
The second part of this question
wants us to calculate an estimate for the mean number of words. And we recall that an estimate for
the mean can be found by dividing the total of the frequency multiplied by midpoint
column by the total frequency. Adding an extra row to our table,
we need to find the totals of the second and fourth columns. Adding the frequencies gives us a
total of 160. And adding the numbers in the last
column for the frequencies multiplied by the midpoints gives us 3391. Our estimate for the mean is
therefore equal to 3391 divided by 160. This is equal to 21.19375. And as we are asked to give our
answer to two decimal places, this is equal to 21.19. This is an estimate for the mean
number of words per sentence in the book.
