I am confused on problem b. The textbook states that the lower quartile is in interval 36-43, but to me, that doesn't make sense because since the lower quartile range of 14 numbers is between the 7th and 8th number, which is from interval 36-43 AND interval 44-51.
On
One definition is of the interval containing the lower quartile is that not more than 25% of observations fall in intervals below and nor more than 75% of them fall in intervals above.
You have $2/28 = 14.3\% \le 25\%$ below interval 36-43 and $21/28 = 75\%$ above. So interval 36-43 qualifies according to that definition.
However, @HenningMakholm (+1) is correct that there are many different conventions for defining quantiles, including quartiles. Distinctions between these varying definitions become relatively unimportant in very large datasets, but (as you see here) can lead to puzzles for small datasets.
A few frequently-used definitions of the lower quartile could put it somewhere between 43 and 44 (maybe at 43.5). In that case, on a technicality, no interval would contain the lower quartile. Details would depend on exactly where within the interval 36-43 the observations actually happen to lie. (Definitions can get even stranger when there are tied numbers near the quartiles.)
Remarkably, I have never seen an elementary statistics book that admits its definition of quartiles is only one of several possible definitions, and that other tests and various software programs use slightly different conventions.
My recommendation: (a) While in a course using this book, cheerfully try to use whatever definition it chooses. (b) Remember the basic idea that roughly speaking quartiles divide the sorted data into four 'chunks' of approximately equal size. (c) Then in any future statistical work use whatever definition is convenient.
Note: R statistical software allows the user to choose among nine 'types'
of quantiles. Here is a dataset that roughly matches the one in your Question,
along with output of quantiles for three of the nine types. (The quantile procedure prints quartiles, unless something else in requested.)
x = c(29,31,37,28,39,42,42,44,45,46,47,48,48,49,50,53,54,55,56,58,60,61,62,63,67,67,68,72)
quantile(x, type=1)
0% 25% 50% 75% 100%
28 42 49 60 72
quantile(x, type=2)
0% 25% 50% 75% 100%
28.0 43.0 49.5 60.5 72.0
quantile(x, type=6)
0% 25% 50% 75% 100%
28.00 42.50 49.50 60.75 72.00
It's either a badly constructed exercise, or the book is trying to teach a particular convention for what "quartile" should mean when there are a multiple of $4$ samples.
Before discounting it completely, look to see if it doesn't have an exact definition that you can apply to the case.