To determine the population variance, we first need to understand what the term "population variance" means.
The population variance, often denoted by [tex]\(\sigma^2\)[/tex], is a measure of the dispersion or spread of a set of values in an entire population. It's calculated from every member of the population.
Now, in the context of your question, let's break down the information given:
- We have a total population of 508 students.
- A sample of 69 students is taken from this population.
When calculating population variance, we incorporate the values from the entire population, not just a sample. This means we are interested in the variance that considers all 508 students. This is critical because any variance calculated from a subset (sample) would be considered the sample variance and not representative of the entire population's variance.
From the options provided:
A. 508 - This is the total population size, which is what we use when referring to the population variance.
B. Neither 69 nor 508 - This option is incorrect since the population variance pertains to the whole population of 508 students.
C. Both 69 and 508 - This is incorrect because the population variance cannot pertain to both the sample size and the population size simultaneously.
D. 69 - This pertains to the sample size, and sample variance uses the sample size, not the population variance.
Therefore, the correct answer is:
A. 508
This means that the population variance [tex]\(\sigma^2\)[/tex] is the variance of the entire population of 508 students.
