A sample of size 300 is to be taken at random from an infinite population. Given that the population proportion is 0.70, the probability that the sample proportion will be smaller than 0.75 is:
A)0.9706.
B)0.4772.
C)0.4706.
D)0.9772.

Question

Quizplus · Accepted Answer

The sample proportion ($\hat{p}$) follows a normal distribution with mean $\mu_{\hat{p}} = p$ and standard deviation $\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}$, where $p = 0.70$ is the population proportion and $n = 300$ is the sample size. The Z-score for a sample proportion of 0.75 is calculated as $Z = \frac{\hat{p} - p}{\sigma_{\hat{p}}}$. Substituting the values, we find the Z-score and then use the standard normal distribution to find the probability that $\hat{p} < 0.75$, which is approximately 0.9706.

A Sample of Size 300 Is to Be Taken at Random