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I am confused with the answer given in the A-Level book S1. Please see the following question and please tell me why answer is 38 and not 24. Is it misprinting or am I missing out something?

Question: The nurse at a health centre records the heights, h cm, to the nearest cm, of a group of boys in the same school age group. The frequency table shows the result

\begin{array}{|c|c|c|} \hline h& Frequency & Cumulative freqency \\ \hline 165& 8& 8&\\ \hline 166& 7& 15&\\ \hline 167& 9& 24&\\ \hline 168& 14& 38&\\ \hline 169& 18& 56&\\ \hline 170& 16& 72&\\ \hline \end{array}.

State the number of boys who are less than 168 cm tall.

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Usually, "less than" implies "less than but not equal to". Otherwise, the explicit "less than or equal to" is used. –  Alfonso Fernandez Dec 29 '12 at 0:02
    
so answer is 8 + 7 + 9 = 24. but book gives answer 38. So I am right to say that 24 boys have height less than 168cm. –  user18096 Dec 29 '12 at 0:03
    
Oh, I read it wrong... If the book included 168, then either the book is wrong, or the conventions in the A-levels (they're British, right?) are different. Are you sure the original wasn't "at most 168cm" or something like that? –  Alfonso Fernandez Dec 29 '12 at 0:14
    
No I have copied exactly what is written in A-level (yes, they're British) book. I am pretty sure it is misprinting as it doesn't make sense to me. I just wanted to double-check. –  user18096 Dec 29 '12 at 0:18
    
I would still check other places in the book or ask a teacher to make sure this is not used consistently, but you're most probably fine. Good luck with you A-levels! –  Alfonso Fernandez Dec 29 '12 at 0:22

1 Answer 1

Less than $168$ presumably includes numbers like $167.92$. Since $168$ (exactly) is unlikely, the answer $38$ is substantially more reasonable than $24$. Less than $168$ does not mean $167$ or below.

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This is true if you interpret "to the closest integer" as "to the closest larger integer". If you interpret "to the closest lower integer", the opposite is true, and if you interpret "to the actual closest integer" then you get no information and are forced to assume that the highschool textbook doesn't deal with that subtlety... –  Alfonso Fernandez Dec 29 '12 at 0:25
    
As entries in column h are rounded to nearest cm, so you are treating entries in column h as 'class' with class boundary 167.5-168.5 for fourth entry (which is 168). Is that right? If that's so then it makes a 'bit' sense. Though still what if some boys height is 168.3. That boy is included in this class. So it doesn't make complete sense as answer is quite dodgy. –  user18096 Dec 29 '12 at 0:28

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