Answer:
The sampling distribution of [tex]\hat p[/tex] is: [tex]\hat p\sim N(p,\ \frac{p(1-p)}{n})[/tex].
Step-by-step explanation:
According to the Central limit theorem, if from an unknown population large samples of sizes n > 30, are selected and the sample proportion for each sample is computed then the sampling distribution of sample proportion follows a Normal distribution.
The mean of this sampling distribution of sample proportion is:
[tex]\mu_{\hat p}=p[/tex]
The standard deviation of this sampling distribution of sample proportion is:
[tex]\sigma_{\hat p}=\sqrt{\frac{p(1-p)}{n}}[/tex]
The study was conducted using the data from 15,000 students.
Since the sample size is so large, i.e. n = 15000 > 30, the central limit theorem is applicable to approximate the sampling distribution of sample proportions.
So, the sampling distribution of [tex]\hat p[/tex] is: [tex]\hat p\sim N(p,\ \frac{p(1-p)}{n})[/tex].