Question

Hi,

Working through a chapter on sampling distributions and getting stuck very early on something that must be so simple the book didn't see fit to explain properly.

Example: In a certain population 30% of people are of blood type A. A random sample of size 5 is drawn. Therefore the population proportion with type A is p = 0.3.
The possible values of pˆ are 0, 0.2, 0.4, 0.6, 0.8, 1.

Why are these the possible values of p^? The numbers given create 5 intervals - is it because we have a sample of 5? If the sample was 10, would the possible values changed? It makes sense to me that they would, but my book doesn't go this way, the next paragraph shows that P(0.2 ≤ pˆ ≤ 0.4) increases as the sample size increases so it seems that the 'possible values' haven't moved.

Could anybody explain to me what is the story with the 'possible values' pf p^ here?

Thanks a lot

Answer 1

Sample proporton is the fraction of the sample which gives favourable outcome. It is the proportion of individuals in a sample which share common traits. For example, for example given in the question, the sample proportion will be the proportion of people of blood type A in a given sample.

Mathematically it is defined as

P^ = Number of favourable outcome in the sample / sample size.

Our sample size is given as 5, therefore sample proportion will be = Number of favourable outcome in the sample / 5

Since the sample size is 5, therefore out of those 5 people there can be either 0, 1, 2, 3, 4 or 5 people of blood type A. Hence P^ takes the values 0, 0.2, 0.4, 0.6, 0.8, 1

P^ = 0/5 = 0    (If 0 out of 5 have blood type A)
P^ = 1/5 = 0.2       (If 1 out of 5 have blood type A)
P^ = 2/5 = 0.4       (If 2 out of 5 have blood type A)
P^ = 3/5 = 0.6 (If 3 out of 5 have blood type A)
P^ = 4/5 = 0.8 (If 4 out of 5 have blood type A)
P^ = 5/5 = 1          (If 5 out of 5 have blood type A)

This is why he possible values of P^ are 0, 0.2, 0.4, 0.6, 0.8, 1.

Similarly, if the sample size was 10, the possible values P^ can take is 0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.

The paragraph P(0.2 ≤ pˆ ≤ 0.4) means the probability when the sample proportion P^ in between 0.2 and 0.4, which infact will increases when the sample size increases