Give the correct choices of these multiple choice questions in questions (a) and (b) and explain your choices (for example: why quantitative and not q

Step 1 
Data in statistics can be divided into two categories, such as quantitative data and qualitative data, depending on how they are measured.
Quantitative variable can take numerical values, whereas qualitative variable is used to identify the characteristics of a variable as categories. Moreover, mathematical operations cannot be performed on qualitative data, but can be performed on quantitative data. 
The continuous variable is a type of quantitative variable that can take any value within two admissible values. The discrete random variable takes a finite or countable number of values. 
Levels of measurement: 
Nominal: Arithmetic operations cannot be performed even if the values are numerical when the possible observations are tags, categories, or names without any natural ordering.
Ordinal: When the possible observations are tags or categories or names, with a natural ordering, the level of measurement is ordinal, arithmetic operations cannot be performed even if the values are numerical. 
Interval: When the possible observations are numerical values that can be added or subtracted, but not divided or multiplied, the level of measurement is interval, zero is not absolute, but relative. 
Ratio: When the possible observations are numerical values that can be added or subtracted, divided, or multiplied, the level of measurement is ratio, zero is absolute. 
Step 2 
a. 
Birth year of family is a discretized measure of time. It measures on numerical scale. Moreover, any mathematical operations can be performed on this. Hence, it is a quantitative variable. 
Birth year us finite and it can take finite number of values. Hence, it is discrete variable. 
The possible observations of birth years are numerical values that can be added or subtracted. However, it cannot be divided or multiplied. In addition, a zero birth year is not absolute. Thus, the level of measurement is interval. 
b. 
The gender can not take numerical values and it is used to categorized the sex of a person. Any mathematical operations cannot be performed on this. Hence, it is a qualitative data. 
As it is a qualitative data, it can not take any values of measurement. Hence, it is neither discrete nor continuous. 
The possible observations of gender are male, female or other, without any natural ordering, the level of measurement is nominal.