Problem 1
No, it is not a statistical question
Explanation:
This is because the question is directed at a specific family, your family in particular. If the surveyor is only asking one person, then there's no need for statistics since a sample isn't formed.
If the question is posed to many people taking a survey, then it would be statistical in nature. However, no further context or clarification is given, so I'll assume that the question is directed at one person, or one family, only. I think your teacher would say "During which month do families take vacation?" to be more general and involve statistics.
Problem 2
The statistical variable is the number of tv sets owned, and this variable is quantitative
Explanation:
The variable is what we want to figure out, and it's random in nature. Hence the name "random variable". It takes on nonnegative whole numbers 0,1,2,3,... so we can see the data set is discrete. The term "discrete" in math means that there aren't infinitely many numbers between two values. For example, if we pick 2 and 5, we can see the values 3,4 are between them and that's it. Contrast this with a continuous data set and there are infinitely many real numbers between 2 and 5 (eg: 2.78912 and 3.412)
While it might be tempting to think of a discrete number set as finite, that's not entirely true. We have infinitely many nonnegative whole numbers to work with. Though of course, there's a realistic limit that we arrive to and someone having 277 televisions is completely out of the ordinary, if not impossible.
The data variable "Number of TV sets owned" is not only discrete, but it's also ordinal, interval and ratio data as well. Here's a breakdown of what I mean
Ordinal = The data values 0,1,2,3,... can be ordered and it makes sense to do so. Usually the order is from smallest to largest.Interval = Gaps between any two values make sense. So going back to 2 and 5, the gap here is 3. Meaning that if family A has 2 tvs and family B has 5 tvs, then we can say that family B has 3 more tvs compared to family A. Such a statement is valid which means interval data applies.Ratio = Not only can we subtract, but we can also divide. If family A has 2 tvs and family B has 6 tvs, then we can see that the second family has 6/2 = 3 times as many televisions compared to the first family. Again, you have to ask if such a statement makes sense for this type of data to apply.
Since we have many terms apply to this, it sometimes helps to pick just one and try to have it describe the others implicitly. So we go for ratio data since that encapsulates interval and ordinal data types as well. If something is ratio, then automatically it must be ordinal and interval. Division depends on the numbers being well ordered, and that gaps between values make sense and are consistent.
To recap, the data type is quantitative, discrete, and we have a ratio scale.
The variable "number of tv sets owned" is not categorical since the choices to pick from 0,1,2,3,... aren't names; instead they are numbers in which we can perform math operations on and have it make sense. So that's why we go for quantitative data instead of categorical. If it asked "what brand of TV are owned by families?", then the categories of the various brand names would be the answer choices, and lead to categorical data.
Side note: categorical data is always qualitative
